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LLM Evaluation as Tensor Completion: Low Rank Structure and Semiparametric Efficiency

Jiachun Li, David Simchi-Levi, Will Wei Sun

Year: 2026Venue: arXiv preprintArea: stat.METype: PreprintEmbeddings: 418

Intelligence

Status: succeeded | Model: google/gemini-3.1-flash-lite-preview | Prompt: intel-v1 | Confidence: 94%

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Summary

The paper presents a semiparametric framework for evaluating Large Language Models (LLMs) by treating leaderboard construction as a low-rank tensor completion problem. It addresses the challenges of noisy, sparse, and non-uniform pairwise human judgments using Bradley-Terry-Luce (BTL) models. The authors derive efficient influence functions and semiparametric efficiency bounds, introducing a 'score-whitening' method to handle anisotropic information geometry and ensure stable inference for both linear and nonlinear functionals of the latent score tensor.

Entities (5)

Bradley-Terry-Luce Model · statistical-model · 99%LLM evaluation · task · 95%Score-whitening · technique · 95%Tensor Completion · methodology · 95%Semiparametric Efficiency · theoretical-concept · 92%

Relation Signals (3)

Bradley-Terry-Luce Model → usedin → LLM evaluation

confidence 98% · We model preference outcomes using Bradley-Terry-Luce-type comparison models.

LLM evaluation → modeledas → Tensor Completion

confidence 95% · This places LLM evaluation in a new tensor completion setting.

Score-whitening → improves → Inference Stability

confidence 92% · We introduce a score-whitening method that... restores stable inference.

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Abstract

Abstract:Large language model (LLM) evaluation platforms increasingly rely on pairwise human judgments. These data are noisy, sparse, and non-uniform, yet leaderboards are reported with limited uncertainty quantification. We study this as semiparametric inference for a low-rank latent score tensor observed through pairwise comparisons under Bradley-Terry-Luce-type models. This places LLM evaluation in a new tensor completion setting with structured observations, non-uniform sampling, and pairwise contrasts. Our target is a smooth functional $\psi(T^\star)$, including linear estimands such as ability gaps and nonlinear ones such as win probabilities. We derive the information operator on the low-rank tangent space, the efficient influence function, and the semiparametric efficiency bound, then construct a one-step debiased estimator with asymptotic normality. A central challenge is that the information operator is anisotropic and does not commute with the tangent-space projection, creating a bottleneck absent from isotropic models. We introduce a score-whitening method that equalizes local Fisher information and restores stable inference at the optimal sample-complexity scale. Our results provide a principled framework for uncertainty quantification in LLM evaluation and more broadly for inference on low-rank structures from pairwise data.

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LLM Evaluation as Tensor Completion: Low-Rank Structure and Semiparametric Efficiency Jiachun Li jiach334@mit.edu David Simchi-Levi dslevi@mit.edu Will Wei Sun sun244@purdue.edu Massachusetts Institute of Technology Purdue University Abstract Large language model (LLM) evaluation platforms increasingly rely on pairwise human judgments, where a user compares two model responses and selects the preferred one. These data are noisy, sparse, and highly non-uniform, yet the resulting leaderboards are typically reported with limited uncertainty quantification. We study this problem as semiparametric inference for a low-rank latent score tensor observed through pairwise comparisons. Specifically, we model preference outcomes using Bradley-Terry-Luce-type comparison models and represent model performance across task categories and other contexts by a latent score tensor T⋆T . This formulation places modern LLM evaluation in a new low-rank tensor completion setting, where the observation pattern is highly structured, the sampling is non-uniform, and each sample reveals only a pairwise contrast rather than a direct tensor entry. Our inference target is a smooth functional ψ(T⋆)ψ(T ), including both linear estimands such as ability gaps and category-specific contrasts and nonlinear estimands such as average win probabilities against a reference pool. We derive the information operator on the identifiable low-rank tangent space, the associated efficient influence function, and the resulting semiparametric efficiency bound. We then construct a one-step debiased estimator with asymptotical normality. A central technical challenge is that, under heterogeneous matchups and non-uniform sampling, the information operator is anisotropic and generally does not commute with projection onto the low-rank tangent space, creating a bottleneck absent from isotropic tensor completion models. To address this, we introduce a score-whitening method that equalizes local Fisher information across comparisons and restores stable inference at the optimal sample-complexity scale. Our results provide a principled framework for uncertainty quantification in LLM evaluation and more broadly for inference on low-rank latent structures from pairwise comparison data. uncertainty quantification, keywords: [class=MSC] keywords: 1 Introduction Large language model (LLM) evaluation has rapidly become a statistical problem of central importance. Modern foundation models are deployed across a wide range of tasks, updated frequently, and often differ only subtly in quality. As a result, the practical question is no longer merely how to obtain a point estimate of model performance, but how to compare models reliably, quantify uncertainty in those comparisons, and do so under highly irregular observation patterns. Human-preference platforms such as Arena have emerged as a central solution in practice: a user submits a prompt, two anonymous models generate responses side by side, and the user votes for the better response; the resulting pairwise outcomes are then aggregated into a public leaderboard (Chiang et al., 2024). Figure 1 illustrates this pipeline. This approach has become widely popular because it evaluates LLM models with real prompts and real users, rather than through a fixed benchmark alone. ⇒ Figure 1: Left plot is the Arena battle-mode interface. A user submits a prompt, two anonymous LLMs respond side by side, and the user votes on which response is preferred. The right plot is a category-specific Arena leaderboard (Text category). Scores are obtained by aggregating pairwise comparison outcomes, while the vote counts vary substantially across models. This imbalance reflects non-uniform sampling across model pairs. The statistical object underlying such leaderboards is naturally a collection of latent pairwise-comparison strengths. A standard starting point is the Bradley-Terry-Luce (BTL) model, under which the probability that model a beats model b depends on the difference of their latent scores (Bradley and Terry, 1952; Luce, 1959). When evaluation is broken down by task category, prompt type, or user segment, the latent score is no longer a single scalar per model but rather a structured array. For example, if j∈[d1]j∈[d_1] indexes models and t∈[d2]t∈[d_2] indexes task categories, one may posit a latent matrix T⋆∈ℝd1×d2T ^d_1× d_2, where Tj,t⋆T _j,t measures model j’s latent ability on category t. More generally, additional modes can be added for user populations, evaluation criteria, or languages, leading to a latent tensor T⋆∈ℝd1×⋯×dm.T ^d_1×·s× d_m. A single battle then reveals only a noisy comparison between two entries of this latent object: if models a and b are compared, one observes a binary outcome Y∈0,1,ℙ(Y=1∣X,T⋆)=σ(⟨T⋆,X⟩),Y∈\0,1\, (Y=1 X,T )=σ\! ( T ,X ), where σ(z)=1/(1+e−z)σ(z)=1/(1+e^-z) and X is a design tensor encoding the relevant pairwise contrast. This viewpoint connects modern LLM evaluation to matrix and tensor completion with structured missingness, with the important difference that the data consist of pairwise contrasts rather than direct entries. The latent score array T⋆T is the scientific object of interest, but only a small and highly uneven subset of pairwise comparisons is observed. Popular frontier models receive many more battles than weaker or newly released models, and prompt traffic is driven by user interests rather than experimental design. Thus the observation pattern is neither balanced nor uniform. The inferential goals in LLM evaluation go well beyond point ranking. In many applications, one seeks to estimate a specific functional of the latent score tensor, such as an ability gap between two models, a category-specific average score, or an average win probability against a reference pool. A generic linear target takes the form ψ(T⋆)=⟨Γ,T⋆⟩,ψ(T )= ,T , where Γ selects or averages entries of T⋆T . In addition, many practical targets are nonlinear. For a fixed category t and a fixed pair of models (a,b)(a,b), one example is the win probability ψ(T⋆)=σ(Ta,t⋆−Tb,t⋆),ψ(T )=σ\! (T _a,t-T _b,t ), (1.1) which gives the probability that model a is preferred to model b on category t. Other examples include category-weighted leaderboard scores, robust summaries such as tail performance, and calibrated transformations of latent scores into public-facing rating scales. This leads to the main question of the paper: How can we construct statistically efficient estimators and valid confidence intervals for linear and nonlinear functionals of a latent low-rank score tensor when the data consist of noisy pairwise comparisons collected under highly non-uniform sampling? Current leaderboard practice does not address this question. Existing platforms focus primarily on ranking models and reporting model-level uncertainty summaries. For example, Arena uses Bradley–Terry-type ranking models and, more recently, the open-source Arena-Rank package, which implements contextual ranking extensions, reweighting for underrepresented models, and confidence intervals based on asymptotic normality of M-estimators (Chiang et al., 2024; Arena Team, 2025). These methods are useful in practice, but they do not explicitly exploit low-rank structure across models, tasks, and related evaluation dimensions, nor do they address semiparametric efficient inference for structured functionals under identification constraints and non-uniform sampling. Low-rank structure provides a natural way to make this problem statistically tractable. In LLM evaluation, it is unrealistic to treat every model-by-task performance cell as unrelated. Rather, performance is typically driven by a small number of latent factors, such as reasoning ability, coding proficiency, instruction following, multilingual robustness, or style sensitivity. A low-rank matrix or tensor model captures this shared structure, reduces effective dimension, and allows information to be borrowed across models and tasks. This is the same principle that underlies low-rank methods in matrix completion, collaborative filtering, and tensor estimation (Candès and Recht, 2009; Keshavan, Montanari and Oh, 2010; Negahban and Wainwright, 2012; Koltchinskii, Lounici and Tsybakov, 2011; Cai et al., 2022; Zhang et al., 2025). In the present setting, it turns a collection of sparse pairwise battles into a coherent structured inference problem. Meanwhile, LLM evaluation is substantially harder than standard completion problems. We observe binary pairwise comparisons rather than entries of T⋆T , so the model is only identifiable up to additive shifts and requires normalization. The observation model is logistic rather than Gaussian, which makes the information carried by each comparison depend on the local Fisher information I(η)=σ(η)1−σ(η)I(η)=σ(η)\1-σ(η)\: closely matched models are informative, while lopsided comparisons are not. The sampling design is also highly uneven across prompts, categories, and model pairs, leading to anisotropic information geometry. Finally, the inferential target is typically not full recovery of T⋆T , but efficient estimation of a functional ψ(T⋆)ψ(T ). These place the problem naturally in the realm of semiparametric inference for structured high-dimensional parameters and motivate the framework developed in this paper. 1.1 Our contributions We study semiparametric inference for low-rank latent score tensors observed through pairwise comparisons. Our contributions are fourfold. A statistical formulation of low-rank LLM evaluation. We formulate leaderboard inference as estimation of a smooth functional ψ(T⋆)ψ(T ) of a low-rank latent score tensor T⋆T under generalized pairwise-comparison models. This unifies linear targets, such as ability gaps and category-specific scores, and nonlinear targets, such as average win probabilities, within a single framework. A key modeling point is that the observed battles correspond to a highly incomplete and non-uniformly sampled collection of pairwise contrasts, so the problem is naturally connected to tensor completion under general missingness mechanisms. Semiparametric efficiency bounds via an information equation. For smooth targets ψ(T⋆)ψ(T ), we derive the efficient influence function (EIF) and the corresponding semiparametric efficiency bound. For a linear functional ψ(T)=⟨Γ,T⟩ψ(T)= ,T , the EIF direction is characterized by the operator equation AH⋆=PΓ,AH =P_T , A=PGP,A=P_TGP_T, where T is the tangent space of the low-rank model under the identification constraint, P_T is the tangent-space projection, and G is the Fisher-information operator induced by the pairwise-comparison model and the sampling distribution. The resulting variance bound Veff(ψ)=⟨PΓ,A−1PΓ⟩V_eff(ψ)= P_T ,\,A^-1P_T plays the role of a generalized Cramér–Rao bound for structured comparison-based inference. To our knowledge, this is the first such efficiency characterization tailored to low-rank pairwise-comparison models with identification constraints and general non-uniform sampling. One-step inference and a new bottleneck. We construct a one-step debiased estimator based on the efficient influence function. Its analysis reveals a new difficulty: under heterogeneous pairwise information, the Fisher-information operator G generally does not commute with the tangent-space projector P_T. As a result, the restricted inverse (PGP)−1(P_TGP_T)^-1 may exhibit unfavorable entrywise amplification, creating a dimension-dependent bottleneck for standard debiasing arguments. This issue is absent in isotropic completion settings where the information operator is proportional to the identity. Score whitening, inverse-probability weighting, and nonlinear targets. To overcome this, we introduce a score-whitening method that normalizes the score by the local Fisher information and restores an isotropic effective information operator on the tangent space. This can be viewed as a semiparametric preconditioning. The resulting estimator attains asymptotic normality at the optimal sample-complexity scale even when the original information geometry is highly heterogeneous. We further combine this idea with inverse-probability weighting to handle non-uniform sampling and extend the analysis from linear to nonlinear targets. Broader implications for tensor completion. Although our main focus is LLM evaluation, the framework also applies more broadly to low-rank inference problems with heterogeneous information geometry, including tensor completion with heteroscedastic noise, non-uniform sampling, and 1-bit observations. We develop these detailed results in Appendix G. 1.2 Closely related literature Our work is related to two fields: low-rank matrix and tensor inference, and statistical modeling of LLM evaluation through pairwise comparisons. Low-rank matrix and tensor completion. Classical work studies recovery of a low-rank matrix from uniformly sampled entries (Candès and Recht, 2009; Keshavan, Montanari and Oh, 2010; Negahban and Wainwright, 2012; Koltchinskii, Lounici and Tsybakov, 2011). A parallel tensor literature develops analogous methods for low-rank tensors under structured decompositions such as CP and Tucker models (Kolda and Bader, 2009; Cai et al., 2022; Ma and Xia, 2024). More recent work considers non-uniform observation schemes, including covariate-assisted weighting, inverse-propensity correction, and missing-not-at-random mechanisms in matrix and tensor completion (Mao, Chen and Wong, 2019; Mao, Wang and Yang, 2023; Chao, Huang and Needell, 2021; Zhang et al., 2025; Duan et al., 2025). Our setting differs in two key respects: the data consist of pairwise contrasts rather than direct entries, and the goal is not only recovery of the latent low-rank object, but semiparametric efficient inference for smooth functionals under pairwise observations and identification constraints. LLM evaluation and BTL models. A rapidly growing empirical literature studies LLM evaluation through human preferences, pairwise battles, benchmark design, and AI judges (Chiang et al., 2024; Li et al., 2024; Petrova, Gordon and Blindow, 2026; Dong et al., 2026). The Arena line of work established pairwise crowdsourced evaluation as a credible public ranking mechanism (Chiang et al., 2024; Li et al., 2024). At the same time, recent studies point to vulnerabilities and heterogeneities in leaderboard data, including demographic variation, benchmark misalignment, and the sensitivity of rankings to data-collection choices (Singh et al., 2025; Petrova, Gordon and Blindow, 2026). This empirical literature makes the statistical need for principled uncertainty quantification especially clear. Recent theory has made substantial progress on estimation and uncertainty quantification in BTL-type models for pairwise preference data (Gao, Shen and Zhang, 2023; Fan, Hou and Yu, 2024; Fan, Kwon and Zhu, 2025; Fan et al., 2026). Closely related ideas also appear in LLM alignment, where BTL-type preference models underlie modern preference-optimization methods and their robust variants (Ouyang et al., 2022; Bose et al., 2025; Xu et al., 2025; Su, 2026). Related semiparametric ideas have also begun to appear in LLM evaluation; for example, Dong et al. (2026) use auxiliary pairwise comparison signals to improve efficiency in mathematical reasoning evaluation. Our work is complementary but distinct. We study direct pairwise-comparison data generated from a latent low-rank score tensor and develop semiparametric efficiency theory for structured linear and nonlinear functionals of that tensor. Once task categories, user groups, or other contextual dimensions are introduced, the inferential target becomes structured and high dimensional, and low-rank regularity becomes essential for statistical efficiency. In this sense, our paper provides an efficient inferential framework tailored to the sparse, structured, and non-uniform pairwise-comparison data that arise in modern LLM evaluation. 1.3 Organization The paper is organized as follows. Section 2 introduces the low-rank pairwise-comparison model for LLM evaluation. Section 3 develops the semiparametric efficiency theory, including the information equation, efficient influence function, and efficiency bound. Section 4 studies the one-step estimator and the non-commutativity bottleneck under heterogeneous information. Section 5 presents score whitening, inverse-probability weighting, and the extension to nonlinear functionals. Section 6 reports synthetic and real-data experiments. Proofs and additional technical details are deferred to the appendices. 2 Model setting We now formalize the statistical model underlying pairwise evaluation platforms such as Arena; see Figure 1. In a typical interaction, each observation is not a direct measurement of a model’s latent quality, but a pairwise comparison between two models under a particular evaluation context such as task category, evaluation criterion, language, or time period. We begin with the simplest and most concrete formulation, which will serve as the primary running example throughout the paper. Example 2.1 (Model-topic pairwise comparisons via BTL model). Suppose a platform evaluates d1d_1 LLM models across d2d_2 task categories, such as coding, mathematics, creative writing, and multilingual tasks. Let T⋆∈ℝd1×d2T ^d_1× d_2 be a latent score matrix, where Tj,u⋆T _j,u represents the latent ability of model j on task category u. At evaluation round i, the platform observes a task category uiu_i, selects two models pip_i and qiq_i, and records a binary outcome Y(i)∈0,1,Y^(i)∈\0,1\, where Y(i)=1Y^(i)=1 means that model pip_i is preferred to model qiq_i on category uiu_i. The key point is that the outcome depends on the latent scores only through their difference, ηi=Tpi,ui⋆−Tqi,ui⋆. _i=T _p_i,u_i-T _q_i,u_i. Under the Bradley-Terry-Luce (BTL) model, ℙ(Y(i)=1∣pi,qi,ui,T⋆)=σ(ηi)=exp⁡(ηi)1+exp⁡(ηi).P (Y^(i)=1 p_i,q_i,u_i,T )=σ( _i)= ( _i)1+ ( _i). (2.1) Equivalently, if eje_j denotes the jjth standard basis vector, we may write the design matrix as X(i)=(epi−eqi)eui⊤,X^(i)=(e_p_i-e_q_i)e_u_i , so that ηi=⟨T⋆,X(i)⟩. _i= T ,X^(i) . Thus each battle reveals one noisy contrast of the latent score matrix rather than one matrix entry. This formulation is directly connected to the Arena example in Figure 1. The left panel corresponds to the data-collection mechanism and the right panel corresponds to the statistical aggregation step to estimate latent model strengths and produce a leaderboard. In the matrix formulation above, the leaderboard is driven by the latent matrix T⋆T , while each observed battle reveals only one binary comparison involving two of its entries. A first important feature of the BTL model is identifiability. Because only score differences affect the feedback, the latent matrix is not identifiable without normalization. Indeed, for any vector c∈ℝd2c ^d_2, replacing T⋆T by T⋆+d1c⊤T +1_d_1c does not change any difference Tp,u⋆−Tq,u⋆T _p,u-T _q,u and therefore leaves all pairwise probabilities unchanged. To fix a unique representative in each equivalence class, we impose a linear identification constraint. The canonical choice is the zero-sum normalization along the model dimension, d1⊤T⋆=0,1_d_1 T =0, (2.2) meaning that, within each task category, the model scores sum to zero. This constraint will play an essential role in the semiparametric formulation developed later. The BTL model in (2.1) is only one instance of a broader pairwise-feedback framework. More generally, let the latent signal live in the space of real matrices or tensors T⋆∈ℝd1×⋯×dmT ^d_1×·s× d_m, equipped with the trace inner product ⟨U,V⟩=∑j1,…,jmUj1…jmVj1…jm U,V = _j_1,…,j_mU_j_1… j_mV_j_1… j_m. We observe independent pairs (X(i),Y(i))∈ℝd1×⋯×dm×,i=1,…,n,(X^(i),Y^(i)) ^d_1×·s× d_m×Y, i=1,…,n, where the design X(i)X^(i) is drawn from a distribution Π⋆ and, conditional on X(i)X^(i), the observation Y(i)Y^(i) depends on T⋆T only through the scalar index ηi=⟨T⋆,X(i)⟩. _i= T ,X^(i) . That is, the conditional log-likelihood takes the single-index form log⁡p(Y(i)∣X(i),T⋆)=ℓ(Y(i),⟨T⋆,X(i)⟩) p (Y^(i) X^(i),T )= (Y^(i), T ,X^(i) ) (2.3) for a known function ℓ:×ℝ→ℝ :Y×R . The corresponding empirical loss is ℒn(T)=1n∑i=1nϕ(Y(i),⟨T,X(i)⟩),L_n(T)= 1n _i=1^nφ (Y^(i), T,X^(i) ), where ϕφ is the negative log-likelihood loss associated with ℓ . The BTL model is the special case in which =0,1Y=\0,1\ and ℓ(y,η)=yη−log⁡(1+eη) (y,η)=yη- (1+e^η). In realistic LLM evaluation, performance may vary simultaneously across models, task categories, user groups, languages, evaluation dimensions, or time. This naturally leads to a general tensor formulation. Let T⋆∈ℝd1×⋯×dmT ^d_1×·s× d_m be an order-m latent score tensor. The first mode will typically index models, while the remaining modes index contextual variables. An observed comparison again depends on T⋆T only through a linear contrast ⟨T⋆,X⟩ T ,X , where X is a signed design tensor encoding which two models are compared and under which context. As in the matrix case, only certain contrasts of T⋆T are observable, so the tensor is identifiable only after imposing suitable linear constraints. We therefore assume that T⋆T satisfies (T⋆)=0,C(T )=0, (2.4) where C is a known linear operator. In the pairwise-comparison setting, the most important example is again a zero-sum constraint along the model mode, but the formulation in (2.4) allows for more general identification restrictions. To make inference feasible in high dimensions, we assume that T⋆T has low-rank Tucker structure (Kolda and Bader, 2009) T⋆=⋆×1U1×2U2⋯×mUm,T =C ×_1U_1×_2U_2·s×_mU_m, (2.5) where Uk∈ℝdk×rkU_k ^d_k× r_k has orthonormal columns and ⋆∈ℝr1×⋯×rmC ^r_1×·s× r_m is a core tensor. In the matrix case m=2m=2, this reduces to the singular value decomposition. The low-rank assumption indicates that LLM’s model performance is driven by a relatively small number of latent factors such as reasoning ability, coding proficiency, instruction following, multilingual robustness, or stylistic preference. Thus low rank is not merely a computational regularizer; it is a structural assumption that allows the model to borrow strength across tasks and contexts. We further assume the standard μ-incoherence condition on T⋆T , which ensures that the singular vectors are not aligned with any single coordinate direction: for each mode j∈[m]j∈[m], maxa∈[dj]⁡‖ea⊤Uj‖22≤μrjdj, _a∈[d_j]\|e_a U_j\|_2^2\;≤\; μ\,r_jd_j, (2.6) for some incoherence parameter μ≥1μ≥ 1. We now collect several dimensional quantities and structural parameters that will be used throughout the paper. Define d¯:=maxj∈[m]⁡dj,d⋆:=∏j=1mdj,D:=∏k=2mdk, d:= _j∈[m]d_j, d := _j=1^md_j, D:= _k=2^md_k, (2.7) so that d¯ d is the largest mode dimension, d⋆d is the total number of tensor entries, and D is the number of contexts (the product of all non-comparison mode sizes). In the matrix case m=2m=2, we have D=d2D=d_2, d¯=max⁡(d1,d2) d= (d_1,d_2), and d⋆=d1d2d =d_1d_2. The rank tuple is r=(r1,…,rm)r=(r_1,…,r_m) and we write r⋆=∏j=1mrjr = _j=1^mr_j for the core tensor size. Let (k)⋆∈ℝrk×∏j≠krjC_(k) ^r_k× _j≠ kr_j denote the mode-k unfolding of the core tensor ⋆C . Following Ma and Xia (2024), we define the maximum and minimum singular values across all mode unfoldings as λmax:=maxk∈[m]⁡‖(k)⋆‖op,λmin:=mink∈[m]⁡σrk((k)⋆), _ := _k∈[m]\|C_(k) \|_op, _ := _k∈[m] _r_k(C_(k) ), (2.8) and the condition number of T⋆T is κ:=λmaxλmin.κ:= _ _ . (2.9) Since the factor matrices UkU_k have orthonormal columns, the Frobenius norm satisfies ‖T⋆‖F=‖⋆‖F\|T \|_F=\|C \|_F, and the singular values of each mode-k unfolding of T⋆T coincide with those of (k)⋆C_(k) . The Frobenius norm is therefore bounded by r⋆λmin≤‖T⋆‖F≤r⋆λmax, r \, _ \;≤\;\|T \|_F\;≤\; r \, _ , (2.10) so that ‖T⋆‖F≍λmax≍κλmin\|T \|_F _ κ\, _ when r and κ are treated as bounded constants. Under the pairwise-comparison observation model, the noise variance for each comparison is determined by the Fisher information I(η)=σ(η)(1−σ(η))I(η)=σ(η)(1-σ(η)), which depends on the local ability gap η=Tp,u⋆−Tq,u⋆η=T _p,u-T _q,u. We assume throughout that T⋆T has bounded entries: ‖T⋆‖∞≤B\|T \|_∞≤ B (2.11) for a constant B>0B>0. This is natural in the LLM evaluation context, where latent ability gaps are bounded. Under (2.11), the pairwise ability gaps satisfy |η|≤2B|η|≤ 2B, and hence the Fisher information is bounded: cB≤I(η)≤CBc_B≤ I(η)≤ C_B for all observed η, where cB,CB>0c_B,C_B>0 depend only on B. That is, the noise level is heteroscedastic but uniformly of constant order. In particular, no explicit noise parameter σ needs to appear in the error bounds. For simplicity, we also assume that the target functional ψ has bounded support: Assumption 2.2 (Bounded support). The functional gradient ∇ψ(T⋆)=Γ∇ψ(T )= has support size |supp(Γ)|≤M|supp( )|≤ M for a constant M, and bounded ℓ1 _1 norm: ‖Γ‖1≤Cψ\| \|_1≤ C_ψ. Our inferential goal is not necessarily full recovery of the entire latent matrix or tensor. In many applications, one is interested in a lower-dimensional summary of T⋆T . As we mentioned in Section 1, the most basic class consists of linear functionals ψ(T⋆)=⟨Γ,T⋆⟩,ψ(T )= ,T , (2.12) where Γ∈ℝd1×⋯×dm ^d_1×·s× d_m is a known direction. This class includes individual ability scores, pairwise contrasts between two models in a fixed category, and averages over subsets of contexts. For example, if one wishes to compare models a and b on task category u, one may choose Γ=(ea−eb)eu⊤ =(e_a-e_b)e_u in the matrix case, so that ψ(T⋆)=Ta,u⋆−Tb,u⋆.ψ(T )=T _a,u-T _b,u. More generally, if ψ is Gateaux differentiable, one may also consider nonlinear targets such as the win probability in (1.1), calibrated leaderboard scores, or robust performance summaries. In this paper, linear functionals will serve as the main starting point of our theory, while nonlinear functionals will be handled later through their local linearization and influence-function representation. In summary, we study a low-rank latent score matrix or tensor T⋆T , subject to linear identification constraints and observed through noisy pairwise comparisons. This formulation captures the basic structure of Arena-style evaluation and provides a unified framework for pairwise-comparison modeling, low-rank estimation, and semiparametric inference. 3 Semiparametric efficiency: score, information, and the efficiency bound We now study the fundamental limit for estimating a target functional from pairwise comparison data. Since the target is a low-dimensional functional of a high-dimensional latent score matrix or tensor, with the sampling distribution Π⋆ acting as a nuisance parameter, the problem is naturally semiparametric. The key objects are the score, the information operator, and the efficient influence function, which together determine the semiparametric efficiency bound. 3.1 Low-rank tangent space and identifiability A first step in semiparametric analysis is to identify the local perturbation directions that are statistically meaningful. In our setting, not every perturbation of T⋆T is admissible. There are two reasons. First, T⋆T is assumed to be low rank. Let rM_r denote the manifold of matrices (or tensors) of rank at most r. Local perturbations should remain within this manifold, at least to first order, which leads to the tangent space of rM_r at T⋆T , denoted by TanT⋆(r)Tan_T (M_r). Informally, this is the linear space of first-order perturbations that preserve the low-rank structure. Second, under pairwise comparisons, T⋆T is identifiable only after normalization. As discussed in Section 2, adding a common constant along the model mode does not change any pairwise probabilities. These unidentifiable directions must therefore be removed by imposing an identification constraint, for example the linear constraint (T⋆)=0C(T )=0 in (2.4). Combining these two restrictions leads to the signal tangent space :=TanT⋆(r)∩Δ∈ℝd1×⋯×dm:(Δ)=0.T:=Tan_T (M_r)∩ \ ^d_1×·s× d_m:C( )=0 \. (3.1) This is the space of local perturbations that are both compatible with the low-rank model and identifiable from the data. In the semiparametric analysis, T plays the role of the effective parameter space. The following matrix example makes this construction concrete. Example 3.1 (Rank-r, first-mode centered matrix). Consider T⋆=UΣV⊤∈ℝd1×d2T =U V ^d_1× d_2 of rank r, with U∈ℝd1×rU ^d_1× r, V∈ℝd2×rV ^d_2× r orthonormal. The tangent space of rM_r at T⋆T consists of matrices Δ=UA⊤+BV⊤ =UA +BV for A∈ℝd2×rA ^d_2× r, B∈ℝd1×rB ^d_1× r. The linear zero-sum constraint d1⊤Δ=01_d_1 =0 reduces to B⊤d1=0B 1_d_1=0 (since U⊤d1=0U 1_d_1=0 by the constraint on T⋆T ). Let Q∈ℝd1×(d1−1)Q ^d_1×(d_1-1) have orthonormal columns spanning d1⟂1_d_1 , so that B=QCB=QC for some C∈ℝ(d1−1)×rC ^(d_1-1)× r. The signal tangent space is then =UA⊤+QCV⊤:A∈ℝd2×r,C∈ℝ(d1−1)×r.T= \UA +QCV :A ^d_2× r,\;C ^(d_1-1)× r \. (3.2) Let ψ:ℝd1×⋯×dm→ℝψ:R^d_1×·s× d_m denote the target of inference, assumed Gateaux differentiable at T⋆T . Then along any admissible direction H∈H , DψT⋆(H)=⟨∇ψ(T⋆),H⟩=⟨P∇ψ(T⋆),H⟩,∀H∈,D _T (H)= ∇ψ(T ),H = P_T∇ψ(T ),H , ∀ H , (3.3) where P_T denotes orthogonal projection onto T. Thus only the component of the gradient lying in the admissible tangent space matters for local inference. For a linear functional ψ(T)=⟨Γ,T⟩ψ(T)= ,T , the gradient is simply ∇ψ(T⋆)=Γ∇ψ(T )= . 3.2 Score and information operator The information operator quantifies how much statistical information different types of comparisons carry about the latent score matrix or tensor. In the LLM evaluation setting, this information is highly heterogeneous. A closely contested matchup, for example, Claude versus GPT-4 on coding tasks, produces an outcome close to a 50/50 comparison and is therefore highly informative about the underlying ability gap. By contrast, a lopsided comparison between a frontier model and a much weaker baseline yields a near-deterministic outcome and contributes relatively little information. The information operator formalizes this heterogeneity and determines how the comparison design and the local noise structure together affect statistical efficiency. We next quantify how much information the observed comparisons carry about perturbations in the direction of T. Recall from Section 2 that the conditional law of Y given X depends on T⋆T only through the scalar index η⋆=⟨T⋆,X⟩η = T ,X . Define the scalar score by sη(Y,η⋆):=∂ηℓ(Y,η)|η=η⋆,η⋆=⟨T⋆,X⟩.s_η(Y,η ):= _η (Y,η) |_η=η , η = T ,X . (3.4) If we perturb the signal along a direction H∈H , that is, consider the submodel Tε=T⋆+εHT_ =T + H, then the corresponding directional score is ∂εlog⁡pTε,Π⋆(X,Y)|ε=0=sη(Y,η⋆)⟨H,X⟩. . ∂ p_T_ , (X,Y) |_ =0=s_η(Y,η )\, H,X . (3.5) This quantity measures the local sensitivity of the log-likelihood to changes in T⋆T along the direction H. A useful way to interpret (3.5) is as a generalized regression score. In ordinary parametric models, the Fisher information describes how precisely a parameter can be estimated. Here the same role is played by an information operator, because the parameter is not a finite-dimensional vector but a matrix or tensor. Let I(η):=[sη(Y,η)2∣η]I(η):=E\! [s_η(Y,η)^2 η ] denote the scalar Fisher information at index value η. We define the information operator G:ℝd1×⋯×dm→ℝd1×⋯×dmG:R^d_1×·s× d_m ^d_1×·s× d_m by ⟨GU,V⟩=⋆[I(η⋆)⟨U,X⟩⟨V,X⟩],U,V∈ℝd1×⋯×dm, GU,V =E \! [I(η )\, U,X V,X ], U,V ^d_1×·s× d_m, (3.6) where η⋆=⟨T⋆,X⟩η = T ,X . This operator is the natural analogue of a Fisher information matrix in the present high-dimensional setting. The operator G captures two distinct sources of heterogeneity. First, the comparison design may be highly uneven, so some contrasts ⟨U,X⟩ U,X are observed much more often than others. Second, even among observed comparisons, some are more informative than others because the Fisher information I(η⋆)I(η ) depends on the difficulty of the matchup. In the BTL model, for example, closely matched models are much more informative than obviously unequal ones. The following example makes these objects concrete and illustrates why the restricted operator is invertible under natural conditions. Example 3.2 (Uniform pairwise comparisons, matrix case). Consider the matrix setting of Example 2.1 with T⋆∈ℝd1×d2T ^d_1× d_2, and suppose models and tasks are sampled uniformly: at each round, a task u is drawn uniformly from [d2][d_2] and a model pair (p,q)(p,q) is drawn uniformly from all (d12) d_12 pairs. The design matrix is X=(ep−eq)eu⊤X=(e_p-e_q)e_u , and the scalar index is η=Tp,u⋆−Tq,u⋆η=T _p,u-T _q,u. Under the BTL model, the scalar Fisher information at index η is I(η)=σ(η)(1−σ(η))=eη(1+eη)2.I(η)=σ(η)(1-σ(η))= e^η(1+e^η)^2. For a matrix H∈ℝd1×d2H ^d_1× d_2, we have ⟨H,X⟩=Hp,u−Hq,u H,X =H_p,u-H_q,u. Hence the information operator acts as (GH)j,u=1d2(d12)∑u′=1d2∑p<qI(ηpq,u′)(Hp,u′−Hq,u′)(δj,p−δj,q)δu,u′.(GH)_j,u= 1d_2 d_12 _u =1^d_2 _p<qI( _pq,u )\,(H_p,u -H_q,u )\, ( _j,p- _j,q )\, _u,u . In the simplest case where all models are equally matched (η∗η^* = 0 and hence T⋆=0T =0 due to the zero-sum constraint), the Fisher information is constant I(0)=1/4I(0)=1/4, and G simplifies to (GH)j,u=14(d12)∑q≠j(Hj,u−Hq,u)=14(d12)((d1−1)Hj,u−∑q≠jHq,u).(GH)_j,u= 14 d_12 _q≠ j(H_j,u-H_q,u)= 14 d_12 ((d_1-1)\,H_j,u- _q≠ jH_q,u ). If H satisfies the zero-sum constraint ∑j=1d1Hj,u=0 _j=1^d_1H_j,u=0, then ∑q≠jHq,u=−Hj,u _q≠ jH_q,u=-H_j,u, so (GH)j,u=12(d1−1)Hj,u,(GH)_j,u= 12(d_1-1)\,H_j,u, and G acts as a scalar multiple of the identity: G=12(d1−1)IdG= 12(d_1-1)\,Id. Hence, G is diagonal and invertible on this subspace. Since the signal tangent space T lies within the zero-sum subspace (cf. Example 3.1), the restricted operator A=PGPA=P_TGP_T is also invertible on T. When T⋆≠0T ≠ 0, the Fisher information I(ηpq,u)I( _pq,u) varies across comparisons, making G no longer proportional to the identity. However, G remains diagonally dominant on the zero-sum subspace and the restricted operator A remains invertible, provided that no comparison is completely deterministic (i.e., I(η)>0I(η)>0 for all observed indices). Since only directions in T are statistically relevant, the corresponding restricted information operator is A:=PGP:→.A:=P_TGP_T:T . (3.7) This is the operator that governs local efficiency for the target ψ(T⋆)ψ(T ). Assumption 3.3 (Invertibility on the tangent space). The restricted information operator A is invertible on T: there exists c>0c_T>0 such that ⟨H,AH⟩≥c‖H‖F2,∀H∈. H,AH ≥ c_T\|H\|_F^2, ∀ H . Assumption 3.3 ensures that the data are informative along every admissible direction in the tangent space T. That is, any nonzero perturbation of T⋆T within T must induce a nontrivial change in the distribution of the observed comparisons. Under this condition, the optimal local direction for estimating ψ(T⋆)ψ(T ) is characterized by the information equation AH⋆=P∇ψ(T⋆).AH =P_T∇ψ(T ). (3.8) Its unique solution is H⋆=A−1P∇ψ(T⋆)∈.H =A^-1P_T∇ψ(T ) . (3.9) The tensor H⋆H is the analogue of the least favorable direction in semiparametric theory: it is the direction in the parameter space that is hardest for estimating the target functional. Equation (3.8) is best understood as an infinite-dimensional version of a generalized least-squares normal equation. To see the analogy concretely, consider a heteroscedastic linear regression model Yi=xi⊤θ⋆+εiY_i=x_i θ + _i with Var(εi∣xi)=σi2Var( _i x_i)= _i^2. The efficient score for estimating a linear functional γ⊤θ⋆γ θ weights each observation by the inverse noise variance: θ^eff=arg⁡minθ∑i=1nσi−2(Yi−xi⊤θ)2,information matrix: ℐ=∑i=1nσi−2xixi⊤. θ_ eff= _θ _i=1^n _i^-2(Y_i-x_i θ)^2, matrix: I= _i=1^n _i^-2\,x_ix_i . The efficient influence function for γ⊤θ⋆γ θ is ϕi=σi−2(Yi−xi⊤θ⋆)xi⊤ℐ−1γ _i= _i^-2(Y_i-x_i θ )\,x_i I^-1γ, and the optimal direction solves ℐh=γI\,h=γ. In our setting, the scalar inverse-variance weight σi−2 _i^-2 is replaced by the Fisher information I(η⋆)I(η ), the design vectors xix_i become design tensors X, and the information matrix ℐI becomes the operator G. The projection P_T then restricts attention to directions that are both low-rank and identifiable. Thus the information equation says: among all admissible perturbations, the optimal one balances the gradient of the target functional against the information geometry induced by the comparison design. 3.3 Efficient influence function and the efficiency bound The solution H⋆H in (3.9) leads directly to the efficient influence function (EIF), ϕEIF(X,Y)=sη(Y,η⋆)⟨H⋆,X⟩. _EIF(X,Y)=s_η(Y,η )\, H ,X . (3.10) More generally, an influence function may be viewed as the first-order term that captures how an estimator responds to small perturbations in the underlying data-generating distribution. In the semiparametric framework, it provides the leading correction to a plug-in estimator and determines its asymptotic variance. The EIF is the particular influence function that has the smallest asymptotic variance, and hence yields the semiparametric efficiency bound. To see why ϕEIF _EIF is the efficient influence function, it is enough to note two facts. First, it gives the correct first-order representation for the target functional ψ(T⋆)ψ(T ), with the required identity following from the information equation (3.8). Second, it is constructed from a direction H⋆H that lies in the signal tangent space T, so it uses only the part of the data that is informative about T⋆T and is orthogonal to nuisance variation coming from the sampling distribution. These two properties imply that ϕEIF _EIF is the unique influence function with the smallest asymptotic variance. The full semiparametric derivation is given in Appendix B.2. The variance of the EIF is Veff(ψ)=⋆[ϕEIF(X,Y)2]=⟨P∇ψ(T⋆),A−1P∇ψ(T⋆)⟩.V_eff(ψ)=E [ _EIF(X,Y)^2]= P_T∇ψ(T ),A^-1P_T∇ψ(T ) . (3.11) This quantity is the semiparametric efficiency bound. It represents the smallest asymptotic variance achievable by any regular estimator of ψ(T⋆)ψ(T ). To state the lower bound formally, we impose a standard unbiasedness condition. Assumption 3.4 (Global unbiasedness). The estimator ψ^=ψ^((Xi,Yi)i=1n) ψ= ψ((X_i,Y_i)_i=1^n) is unbiased for ψ(T)ψ(T) under the model: T,Π[ψ^]=ψ(T),∀T in the parameter space .E_T, [ ψ]=ψ(T), ∀\,T in the parameter space P. Theorem 3.5 (Information lower bound). Under Assumptions 3.3 and 3.4, any estimator ψ ψ with finite variance satisfies Var⋆(ψ^)≥1nVeff(ψ),Veff(ψ)=⟨P∇ψ(T⋆),A−1P∇ψ(T⋆)⟩.Var ( ψ)≥ 1n\,V_eff(ψ), V_eff(ψ)= P_T∇ψ(T ),A^-1P_T∇ψ(T ) . (3.12) Moreover, the bound is attained by the EIF in the sense that ⋆[ϕEIF(X,Y)2]=Veff(ψ).E [ _EIF(X,Y)^2]=V_eff(ψ). The proof is a standard information-inequality argument and is given in Appendix B.1. The key message of Theorem 3.5 is that once the low-rank geometry, the identification constraint, and the comparison design are taken into account, the efficiency bound takes the same conceptual form as in classical semiparametric theory: the optimal variance is the squared norm of the projected gradient under the inverse restricted information operator. This perspective is especially useful in LLM evaluation because it isolates the precise ways in which the problem differs from standard low-rank completion. In standard matrix/tensor completion with homoscedastic observations and uniform sampling, the information geometry is often isotropic. Here, by contrast, both the observation model and the sampling design may be highly heterogeneous. The restricted operator A=PGPA=P_TGP_T is therefore the correct object for capturing the true precision limit. 4 One-step estimation and upper bound analysis Section 3 characterized the semiparametric efficiency bound through the efficient influence function. We now turn to the constructive question: can one build a feasible estimator that attains this bound in the low-rank pairwise-comparison model? Our approach is based on a one-step correction to an initial low-rank estimator. This is a standard semiparametric strategy in spirit, but its analysis here is substantially more delicate because the data are pairwise, the signal is low rank, and the information operator is generally anisotropic. 4.1 The one-step estimator We use the cross-fitting procedure summarized in Algorithm 1. The data are randomly partitioned into K folds of approximately equal size. For each fold k, the remaining K−1K-1 folds are used to construct the initial estimator and the estimated tangent-space projector, while fold k serves as the evaluation sample for the debiasing step. The final estimate is obtained by averaging over the K folds. Cross-fitting avoids the sample-size loss inherent in a single data split while maintaining the independence between the estimation and evaluation stages that simplifies the theoretical analysis. Algorithm 1 One-step estimator for a linear functional ψ(T⋆)=⟨Γ,T⋆⟩ψ(T )= ,T 1. Randomly partition the data (Xi,Yi)i=1N\(X_i,Y_i)\_i=1^N into K folds 1,…,KD_1,…,D_K of approximately equal size. 2. For each fold k=1,…,Kk=1,…,K: (a) Using the out-of-fold data −k:=⋃j≠kjD_-k:= _j≠ kD_j, compute a low-rank estimator T^(k) T^(k) of T⋆T subject to the identification constraint, and construct the associated tangent-space projector P^(k) P_T^(k). (b) Compute the plug-in information operator G^(k) G^(k) via (4.1) using −kD_-k. (c) Compute the estimated EIF direction H^(k)=(P^(k)G^(k)P^(k))−1P^(k)Γ. H^(k)=( P_T^(k) G^(k) P_T^(k))^-1 P_T^(k) . (d) Using fold k as the evaluation sample, form the fold-level one-step estimator ψ^(k)=ψ(T^(k))+KN∑i∈ksη(Yi,η^i(k))⟨H^(k),Xi⟩,η^i(k):=⟨T^(k),Xi⟩. ψ^(k)=ψ( T^(k))+ KN _i _ks_η(Y_i, η_i^(k))\, H^(k),X_i , η_i^(k):= T^(k),X_i . 3. Output the cross-fitted estimator ψ^=K−1∑k=1Kψ^(k) ψ=K^-1 _k=1^K ψ^(k) and the variance estimate V^=N−1∑i=1Nϕ^i2 V=N^-1 _i=1^N φ_i^2, where ϕ^i=sη(Yi,η^i(ki))⟨H^(ki),Xi⟩ φ_i=s_η(Y_i, η_i^(k_i))\, H^(k_i),X_i and kik_i denotes the fold containing observation i. Let T T be an initial estimator of T⋆T , obtained from an auxiliary sample by a low-rank tensor estimation procedure appropriate for the pairwise-comparison model. In practice, one may use spectral initialization followed by constrained likelihood refinement, or any other estimator that achieves the accuracy conditions stated in Section 4.2. Let P P_T denote the estimated projector onto the tangent space at T T. Since the population information operator G defined in (3.6) depends on the unknown Fisher information I(η⋆)I(η ), it is not directly available. We therefore replace G by its plug-in estimate ⟨G^U,V⟩:=1ninit∑i∈initI(η^i)⟨U,Xi⟩⟨V,Xi⟩,η^i:=⟨T^,Xi⟩, G\,U,V := 1n_ init _i _ initI( η_i)\, U,X_i V,X_i , η_i:= T,X_i , (4.1) where initD_ init denotes the initialization sample and ninit=|init|n_ init=|D_ init|. Under the bounded signal condition (2.11) (which guarantees score regularity; see Appendix G) and the entrywise accuracy of T T, the estimation error G^−G G-G is controlled by standard concentration arguments (see Appendix C.10). We define the estimated efficient-influence-function direction H H by (P^G^P^)H^=P^Γ,that is,H^=(P^G^P^)−1P^Γ.( P_T G P_T) H= P_T , is, H=( P_T G P_T)^-1 P_T . (4.2) For each fold k, the evaluation sample kD_k is independent of the auxiliary quantities T^(k) T^(k) and H^(k) H^(k) constructed from −kD_-k. The fold-level one-step estimator of the linear functional ψ(T⋆)=⟨Γ,T⋆⟩ψ(T )= ,T is ψ^(k)=ψ(T^(k))+KN∑i∈ksη(Yi,η^i(k))⟨H^(k),Xi⟩,η^i(k):=⟨T^(k),Xi⟩. ψ^(k)=ψ( T^(k))+ KN _i _ks_η\! (Y_i, η_i^(k) )\, H^(k),X_i , η_i^(k):= T^(k),X_i . (4.3) Within each fold, the first term is a plug-in estimator, while the second is the influence-function correction. The cross-fitted estimator is the average over all K folds: ψ^=1K∑k=1Kψ^(k). ψ= 1K _k=1^K ψ^(k). (4.4) Intuitively, the fold-level correction removes the leading bias of the plug-in estimator and aligns the resulting estimator with the efficient local direction identified in Section 3. Cross-fitting ensures that the independence between the evaluation sample and the estimated quantities (T^(k),H^(k))( T^(k), H^(k)) holds within each fold, while using the full sample for both estimation and evaluation. For general nonlinear functionals, the same construction applies after replacing Γ by the gradient of the target functional evaluated at the initial estimator. We defer the extension of general nonlinear functionals in Section 5.3. 4.2 Assumptions for the upper-bound analysis To establish asymptotic efficiency of the one-step estimator, we impose three conditions. Assumption 4.1 (Initial estimator). The initial estimator T T satisfies the same column-sum-zero constraint as T⋆T , i.e., ∑a=1d1T^a,u=0 _a=1^d_1 T_a,u=0 for all u∈[D]u∈[D], and has entrywise accuracy ‖T^−T⋆‖∞≤C1d¯logc⁡d¯n,\| T-T \|_∞≤ C_1 d ^c dn, (4.5) for some absolute constants c,C1>0c,C_1>0. The entrywise error bound in Assumption 4.1 is essential for one-step inference, because the score correction is evaluated entry by entry through η^i−ηi⋆=⟨T^−T⋆,Xi⟩, η_i- _i = T-T ,X_i , and in the pairwise-comparison design XiX_i is sparse, so controlling the remainder terms requires uniform control of the local errors in T⋆T . In particular, the score perturbation terms in the proof are governed by maxi⁡|η^i−ηi⋆| _i| η_i- _i |, and hence by ‖T^−T⋆‖∞\| T-T \|_∞. Obtaining (4.5) is substantially more challenging than obtaining a Frobenius bound. In low-rank problems, entrywise control typically requires row-wise perturbation bounds for the estimated singular subspaces. Such bounds are standard under additive noise, but are much harder in generalized linear models and, in particular, under pairwise comparison observations. To address this, we use a row-wise refinement step based on one-step logistic score updates for the factor matrices. Starting from any estimator T^(0) T^(0) with near-optimal Frobenius error, the refinement improves it to an estimator T T satisfying (4.5). The detailed construction of such initial estimator and its guarantees are deferred to Appendix H. Assumption 4.2 (Signal strength). The Frobenius norm satisfies ‖T⋆‖F≥cd⋆\|T \|_F≥ c d (4.6) for an absolute constant c>0c>0. Assumption 4.2 and (2.10) imply that λmin≍d⋆ _ d when κ and r are bounded constants. Since the noise level under the pairwise-comparison model is of constant order (the Fisher information satisfies cB≤I(η)≤1/4c_B≤ I(η)≤ 1/4), the classical signal-to-noise ratio λmin/σ _ /σ grows with d⋆ d , which is the rate required in existing literature when the sample size satisfies n≥C0d¯logc⁡d¯n≥ C_0\, d ^c d (4.7) for a sufficiently large constant C0C_0 depending on μ,κ,r,mμ,κ,r,m. Under this sample-size condition, all σ/λminσ/ _ terms appearing in the subspace perturbation bounds (see Appendix C.1) are absorbed into the structural constants, and all error bounds take the simplified form d¯logc⁡d¯/n d ^c d/n without any explicit signal-to-noise ratio. In the proof, additional subspace-perturbation properties are needed; these follow from Assumptions 4.1–4.2 by standard perturbation theory (Ma and Xia, 2024) and are discussed in Appendix C.1. Assumption 4.3 (Alignment condition). There exists a constant αΓ>0 _ >0 such that ‖PΓ‖F≥αΓd¯1/2(d⋆)−1/2‖Γ‖F.\|P_T \|_F≥ _ \, d^1/2(d )^-1/2\| \|_F. (4.8) Assumption 4.3 requires that the target functional have a non-negligible component in the tangent space T. In many natural examples this condition is mild; for instance, when Γ itself lies in T, as in entrywise contrasts or category-specific ability gaps, the condition holds with αΓ=1 _ =1. 4.3 Main error bound for the one-step estimator We now state the main upper-bound result for the one-step estimator, with leading term given by the efficient influence function and remainder controlled explicitly. Theorem 4.4 (Error bound for the one-step estimator). Under Assumptions 4.1–4.3, the bounded signal condition (2.11), and the sample size condition (4.7), the following holds with probability at least 1−d¯−c1- d^-c conditional on the first-stage sample: |ψ^−ψ(T⋆)−(ℙn−ℙ⋆)ϕ⋆|≤C(μ,κ,r,m)CA‖Γ‖1d¯logc⁡d¯n, | ψ-ψ(T )-(P_n-P )φ |\;≤\;C(μ,κ,r,m)\,C_A\,\| \|_1\, d ^c dn, (4.9) where ϕ⋆φ is the efficient influence function and CAC_A is defined such that ‖A−1‖∞→∞∨‖A^−1‖∞→∞≤CAd⋆, \|A^-1\|_∞→∞\; \;\| A^-1\|_∞→∞\;≤\;C_A\,d , (4.10) whereA:=PGP,A^:=P^G^P^. A:=P_TGP_T, A:= P_T G P_T. Here C(μ,κ,r,m)C(μ,κ,r,m) is a constant depending only on μ,κ,r,mμ,κ,r,m (and the bound B from (2.11)), and c>0c>0 is an absolute constant. Theorem 4.4 gives a sharp, simplified characterization of the one-step estimator in the pairwise-comparison setting. Building on it, we obtain the formal CLT. Theorem 4.5 (CLT and variance estimation for the one-step estimator). Under the conditions of Theorem 4.4, if additionally CAd¯log2c⁡d¯n→ 0,C_A\, d ^2c dn\;→\;0, (4.11) then the one-step estimator is asymptotically normal: n(ψ^−ψ(T⋆))Veff→(0,1), n ( ψ-ψ(T ) ) V_ eff\; d\;N(0,1), (4.12) where the asymptotic variance is the semiparametric efficiency bound Veff=⟨PΓ,A−1PΓ⟩.V_ eff\;=\; P_T ,\,A^-1P_T . (4.13) Moreover, the plug-in variance estimator V^eff:=ℙn[ϕ^2] V_ eff:=P_n[ φ^2], where ϕ^(X,Y):=sη(Y,η^(X))⟨H^,X⟩ φ(X,Y):=s_η(Y, η(X)) H,X , satisfies |V^eff−VeffVeff|≲Cd¯logc⁡d¯n | V_ eff-V_ effV_ eff |\; \;C d ^c dn (4.14) with probability at least 1−d¯−c1- d^-c. The CLT proof proceeds in two steps: the remainder bound (4.9) gives nRn/Veff→0 n\,R_n/ V_ eff→ 0 under (4.11), and the oracle influence-function average satisfies a conditional Lyapunov CLT (Appendix F). The variance estimation bound (4.14) is proved in Appendix C.13; the key observation is that the common d⋆d spectral scale of A−1A^-1 and A^−1 A^-1 cancels in the relative ratio, leaving only the geometric perturbation scale ρ and the alignment factor αΓ _ . The role of CAC_A. The term CAC_A captures the entrywise behavior of the restricted inverse information operator A:=PGP,A:=P_TGP_T, where ⟨H1,G(H2)⟩:=ℙ⋆[I(η⋆)⟨H1,X⟩⟨H2,X⟩] H_1,G(H_2) :=P [I(η ) H_1,X H_2,X ] is the Fisher information operator. The oracle and estimated EIF directions are H⋆:=A−1PΓ,H^:=A^−1P^Γ.H :=A^-1P_T , H:= A^-1 P_T . (4.15) Although the spectral norm of A−1A^-1 is always well-behaved (‖A−1‖op≲d⋆\|A^-1\|_ op d ), the one-step analysis requires the stronger entrywise control ‖A−1‖∞→∞∨‖A^−1‖∞→∞≤CAd⋆.\|A^-1\|_∞→∞\; \;\| A^-1\|_∞→∞\;≤\;C_A\,d . (4.16) The next proposition characterizes the range of CAC_A and identifies when it is dimension-free. Proposition 4.6 (Range of CAC_A). Under the bounded signal condition (2.11) and μ-incoherence: (i) (General bounds.) The term CAC_A satisfies C(μ,r,m)≤CA≤C(μ,r,m)d¯.C(μ,r,m)\;≤\;C_A\;≤\;C(μ,r,m)\, d. (4.17) In the constant-weight benchmark I(η⋆)≡1/4I(η )≡ 1/4, A0−1=2d⋆PA_0^-1=2d P_T and CA=poly(μ,r,m)C_A=poly(μ,r,m). (i) (Near-constant regime.) If ‖T⋆‖∞≤B0\|T \|_∞≤ B_0 for a constant B0=B0(μ,m,r)B_0=B_0(μ,m,r) depending only on the structural parameters, then CA≤C(μ,r,m,B0)C_A≤ C(μ,r,m,B_0): a dimension-free constant. In this regime, the CLT condition (4.11) reduces to d¯log2c⁡d¯/n→0 d ^2c d/n→ 0, and the one-step estimator achieves the semiparametric efficiency bound at the optimal sample-complexity scale n≫d¯polylog(d¯)n d\,polylog( d). The proof is given in Appendix C.12. Key difficulty: non-commutativity of G and P_T. The gap between the O(1)O(1) lower bound and the O(d¯)O( d) upper bound in (4.17) reflects the fundamental technical challenge of this problem: the non-commutativity of the information operator G and the tangent-space projector P_T. In isotropic settings (e.g., additive Gaussian noise), G is proportional to the identity on T, so A−1A^-1 behaves like P_T and CAC_A is automatically dimension-free. In pairwise comparisons, however, the Fisher information I(η⋆)I(η ) varies across matchups, making G anisotropic. Because G and P_T do not commute in general, the inverse A−1=(PGP)−1A^-1=(P_TGP_T)^-1 can have substantially worse entrywise behavior than its spectral norm suggests. This difficulty is unique to our setting. The existing debiasing literature for low-rank matrix and tensor estimation, including work on matrix completion, tensor completion, and nuclear-norm penalized inference, assumes homoscedastic or isotropic designs under which G∝IG I and the operator A does not arise. The quantity CAC_A and its interplay with non-commutative operator inversion are, to our knowledge, entirely new to this line of work. 4.4 Proof outline and technical challenges We briefly explain the structure of the proof of Theorem 4.4 and highlight where the main technical difficulty enters. Let Δ:=T^−T⋆,η^i:=⟨T^,Xi⟩,ηi⋆:=⟨T⋆,Xi⟩. := T-T , η_i:= T,X_i , _i := T ,X_i . Recall that the oracle efficient influence function is ϕ⋆(X,Y)=sη(Y,η⋆)⟨H⋆,X⟩,H⋆=(PGP)−1PΓ.φ (X,Y)=s_η(Y,η ) H ,X , H =(P_TGP_T)^-1P_T . Starting from the one-step estimator (4.4), we add and subtract the oracle correction term and obtain the decomposition ψ^−ψ(T⋆)=(ℙn−ℙ⋆)ϕ⋆⏟leading empirical term+Remp+Rbias, ψ-ψ(T )= (P_n-P )φ _leading empirical term\;+\;R_emp\;+\;R_bias, (4.18) where the empirical remainder RempR_emp and the bias remainder RbiasR_bias are Remp:=(ℙn−ℙ⋆)[ST^(H^)−ST⋆(H⋆)],Rbias:=⟨Γ,Δ⟩+ℙ⋆[ST^(H^)−ST⋆(H⋆)],R_emp:=(P_n-P ) [S_ T( H)-S_T (H ) ], R_bias:= , +P [S_ T( H)-S_T (H ) ], and ST(H)(X,Y):=sη(Y,⟨T,X⟩)⟨H,X⟩.S_T(H)(X,Y):=s_η\! (Y, T,X ) H,X . The first term in (4.18) is the leading stochastic term. It is an average of i.i.d. centered random variables and gives the asymptotically normal limit. The task is therefore to show that the empirical remainder RempR_emp and the bias remainder RbiasR_bias are of smaller order. To do so, we further split the empirical remainder by adding and subtracting ST^(H⋆)S_ T(H ): Remp R_emp =(ℙn−ℙ⋆)[sη(Y,η^)⟨H^−H⋆,X⟩]⏟RempH(direction error) = (P_n-P ) [s_η(Y, η)\, H-H ,X ]_R_emp^H\ (direction error) +(ℙn−ℙ⋆)[(sη(Y,η^)−sη(Y,η⋆))⟨H⋆,X⟩]⏟Rempη(score perturbation), +\; (P_n-P ) [(s_η(Y, η)-s_η(Y,η ))\, H ,X ]_R_emp^η\ (score perturbation), (4.19) where η^:=⟨T^,X⟩ η:= T,X and η⋆:=⟨T⋆,X⟩η := T ,X . Here RempHR_emp^H measures the fluctuation caused by replacing the oracle EIF direction H⋆H with its estimate H H, while RempηR_emp^η captures the additional error from evaluating the score at η η instead of η⋆η . Similarly, we split the bias term by decomposing Γ=PΓ+(I−P)Γ =P_T +(I-P_T) and again adding and subtracting ST^(H⋆)S_ T(H ): Rbias R_bias =⟨(I−P)Γ,Δ⟩⏟Rproj+ℙ⋆[ST^(H^)−ST^(H⋆)]⏟RbiasH = (I-P_T) , _R_proj\;+\; P [S_ T( H)-S_ T(H ) ]_R_bias^H +⟨PΓ,Δ⟩+ℙ⋆[ST^(H⋆)−ST⋆(H⋆)]⏟Rcancel. +\; P_T , +P [S_ T(H )-S_T (H ) ]_R_cancel. (4.20) The first term RprojR_proj is a projection leakage term. The second term RbiasHR_bias^H is the population-level error due to estimating the EIF direction. The final term RcancelR_cancel contains the key first-order cancellation of the one-step estimator. The dominant difficulty is the control of RempHR_emp^H and RbiasHR_bias^H, both of which depend on H^−H⋆ H-H and are thus governed by CAC_A (see the discussion in Section 4.3). The remaining terms are controlled by standard arguments: RempηR_emp^η by smoothness of the score and entrywise accuracy of T T; RprojR_proj by subspace estimation error; and in RcancelR_cancel, the tangent-space plug-in bias cancels with the population correction, leaving only higher-order contributions. See Table 1 for a summary. Term Description Simplified bound Proof RempHR_emp^H direction-error emp. process CA‖Γ‖1d¯logc⁡d¯/nC_A\,\| \|_1\, d ^c d/n App. C.4 RempηR_emp^η score-perturbation emp. process CA‖Γ‖1d¯logc⁡d¯/nC_A\,\| \|_1\, d ^c d/n App. C.5 RprojR_proj projection leakage ‖Γ‖1d¯logc⁡d¯/n\| \|_1\, d ^c d/n App. C.6 RbiasHR_bias^H bias from H^≠H⋆ H≠ H CA‖Γ‖1d¯/nC_A\,\| \|_1\, d/n App. C.7 RcancelR_cancel (1st order) ⟨PΓ,Δ⟩ P_T , cancellation CA‖Γ‖1d¯logc⁡d¯/nC_A\,\| \|_1\, d ^c d/n App. C.8 RcancelR_cancel (2nd order) O(‖Δ‖∞2)O( _∞^2) remainder CA‖Γ‖1d¯/nC_A\,\| \|_1\, d/n App. C.9 CLT remainder nRn n\,R_n vs. Veff V_ eff CAd¯log2c⁡d¯/nC_A d ^2c d/n App. F Table 1: Summary of remainder terms and their simplified bounds after substituting the pairwise-comparison scalings σ=O(1)σ=O(1), λmin≍d⋆ _ d , and δ=d¯−cδ= d^-c. All bounds hold with probability ≥1−d¯−c≥ 1- d^-c. 5 Score whitening and extensions Section 4 showed that the main obstacle to achieving semiparametric efficiency at the optimal sample-complexity scale is the non-commutativity of the information operator G and the tangent-space projector P_T. This obstruction is reflected in the quantity CAC_A, which can become dimension dependent when the local Fisher information varies substantially across matchups. In Section 5.1, we introduce a simple remedy: score whitening. The basic idea is to rescale each comparison by its local Fisher information so that all matchups contribute on a common information scale. This transformation turns the effective information operator into an isotropic one, removes the CAC_A-driven bottleneck, and yields a simpler one-step estimator with stable asymptotic behavior. We then show how the same idea extends to non-uniform sampling through inverse-probability weighting in Section 5.2 and to nonlinear targets through local linearization in Section 5.3. 5.1 The score-whitened one-step estimator In the BTL model, the information carried by a single comparison depends strongly on the underlying ability gap. If two models are closely matched, the comparison is highly informative; if one model dominates the other, the outcome is nearly deterministic and carries little information. As mentioned in Example 3.2, for a comparison with linear index η, the Fisher information is I(η)=σ(η)1−σ(η),σ(η)=(1+e−η)−1.I(η)=σ(η)\1-σ(η)\, σ(η)=(1+e^-η)^-1. Thus the information operator is intrinsically heterogeneous. This heterogeneity is what makes G anisotropic and creates the non-commutativity problem identified in Section 4. The key observation is that this heterogeneity can be removed by re-scaling the score. Given the score function sη(y,η)=∂ηℓ(y,η)s_η(y,η)= _η (y,η), define the whitened score s~η(y,η):=sη(y,η)I(η). s_η(y,η):= s_η(y,η)I(η). (5.1) This whitened score remains centered: ⋆[s~η(Y,η⋆)∣X]=0.E [ s_η(Y,η ) X]=0. Second, its conditional derivative becomes constant: ⋆[∂ηs~η(Y,η⋆)∣X]=−1.E \! [ _η s_η(Y,η ) X ]=-1. This second identity is crucial and means that, after whitening, the Fisher-information weight no longer varies across comparisons. As a consequence, under a uniform pairwise-comparison design, the effective information operator becomes isotropic. By uniform design, we mean that the comparison tensor X is sampled uniformly over all admissible context–pair combinations; for example, in the model–topic setting, this means sampling the task category uniformly and the unordered model pair uniformly from all (d2) d2 possible pairs. Under this design, every valid comparison receives the same sampling probability. More precisely, on the column-sum-zero subspace, G0(H)=⋆[⟨H,X⟩X]=1d⋆H,G_0(H)=E [ H,X X]= 1d H, (5.2) where d⋆d is the effective comparison dimension. Therefore A0=PG0P=1d⋆P,A0−1=d⋆P.A_0=P_TG_0P_T= 1d P_T, A_0^-1=d P_T. This is exactly the isotropic structure that was missing in the general upper-bound analysis shown in Theorem 4.4. In particular, the inverse restricted information operator is now explicit, and the troublesome quantity CAC_A disappears. Under the whitened information structure, the efficient direction simplifies dramatically. The oracle and plug-in directions become Hws⋆=d⋆PΓ,H^ws=d⋆P^Γ.H _ ws=d P_T , H_ ws=d P_T . (5.3) No operator inversion is required; the direction is obtained simply by projecting the target gradient onto the estimated tangent space and multiplying by d⋆d . This is the main computational and theoretical advantage of whitening. The corresponding whitened one-step estimator is ψ^ws=⟨Γ,T^⟩+ℙn[s~η(Y,η^)⟨H^ws,X⟩],η^=⟨T^,X⟩. ψ_ ws= , T +P_n\! [ s_η(Y, η)\, H_ ws,X ], η= T,X . (5.4) Like the general one-step estimator, this is a plug-in estimator plus a first-order correction. The difference is that the correction now uses the whitened score and the simplified direction (5.3). See Algorithm 2 for details. Algorithm 2 Score-whitened one-step estimator for a linear functional ψ(T⋆)=⟨Γ,T⋆⟩ψ(T )= ,T 1. Randomly partition the data (Xi,Yi)i=1N\(X_i,Y_i)\_i=1^N into K folds 1,…,KD_1,…,D_K of approximately equal size. 2. For each fold k=1,…,Kk=1,…,K: (a) Using the out-of-fold data −k:=⋃j≠kjD_-k:= _j≠ kD_j, compute a low-rank estimator T^(k) T^(k) of T⋆T subject to the identification constraint, and construct the associated tangent-space projector P^(k) P_T^(k). (b) Compute the whitened EIF direction H^ws(k)=d⋆P^(k)Γ. H_ ws^(k)=d \, P_T^(k) . (c) Using fold k as the evaluation sample, form the fold-level whitened one-step estimator ψ^ws(k)=ψ(T^(k))+KN∑i∈ksη(Yi,η^i(k))I(η^i(k))⟨H^ws(k),Xi⟩, ψ_ ws^(k)=ψ( T^(k))+ KN _i _k s_η(Y_i, η_i^(k))I( η_i^(k))\, H_ ws^(k),X_i , where η^i(k):=⟨T^(k),Xi⟩ η_i^(k):= T^(k),X_i . 3. Output the cross-fitted estimator ψ^ws=K−1∑k=1Kψ^ws(k) ψ_ ws=K^-1 _k=1^K ψ_ ws^(k) and the variance estimate V^ws=N−1∑i=1Nϕ~^i2 V_ ws=N^-1 _i=1^N φ_i^2, where ϕ~^i=sη(Yi,η^i(ki))/I(η^i(ki))⟨H^ws(ki),Xi⟩ φ_i=\s_η(Y_i, η_i^(k_i))/I( η_i^(k_i))\\, H_ ws^(k_i),X_i and kik_i denotes the fold containing observation i. The next two theorems give the error bound and CLT for the score-whitened estimator, paralleling Theorems 4.4 and 4.5 for the general case. Theorem 5.1 (Error bound for the score-whitened estimator). Under the same conditions as Theorem 4.4 and a uniform sampling design, with probability at least 1−d¯−c1- d^-c, ψ^ws−ψ(T⋆)=(ℙn−ℙ⋆)ϕ~0⋆+Rn, ψ_ ws-ψ(T )=(P_n-P ) φ_0 +R_n, (5.5) where ϕ~0⋆(X,Y):=s~η(Y,η⋆)⟨Hws⋆,X⟩ φ_0 (X,Y):= s_η(Y,η )\, H _ ws,X is the whitened influence function and the remainder satisfies |Rn|≤C(μ,κ,r,m)‖Γ‖1d¯logc⁡d¯n.|R_n|\;≤\;C(μ,κ,r,m)\,\| \|_1\, d ^c dn. (5.6) The bound (5.6) is the direct analogue of (4.9), with the crucial difference that no CAC_A factor appears. The proof follows the same five-term decomposition (Appendix D), but each term simplifies: the direction error is H^ws−Hws⋆=d⋆(P^−P)Γ H_ ws-H _ ws=d ( P_T-P_T) (no operator inversion), and the first-order cancellation is complete (no off-tangent residual), since G0(Hws⋆)=PΓG_0(H _ ws)=P_T . Theorem 5.2 (CLT and variance estimation for the score-whitened estimator). Under the conditions of Theorem 5.1, since n≫d¯logc⁡d¯n d ^c d implies nRn→0 n\,R_n→ 0, the conditional Lyapunov CLT gives n(ψ^ws−ψ(T⋆))Vws→(0,1), n ( ψ_ ws-ψ(T ) ) V_ ws\; d\;N(0,1), (5.7) with asymptotic variance Vws=⋆[⟨d⋆PΓ,X⟩2I(η⋆)].V_ ws=E \! [ d P_T ,X ^2I(η ) ]. (5.8) Also, the plug-in estimator V^ws=ℙn[ϕ~ 2] V_ ws=P_n[ φ^\,2]with ϕ~^(X,Y)=s~η(Y,η^(X))⟨d⋆P^Γ,X⟩ φ(X,Y)= s_η(Y, η(X)) d P_T ,X satisfies |V^ws−VwsVws|≲Cd¯logc⁡d¯n | V_ ws-V_ wsV_ ws |\; \;C d ^c dn (5.9) with probability at least 1−d¯−c1- d^-c. In particular, no CAC_A factor appears in (5.9), because the whitened direction H^ws=d⋆P^Γ H_ ws=d P_T involves no operator inversion. Comparing Theorems 4.5 and 5.2 reveals a clear efficiency–robustness trade-off. The general one-step estimator targets the semiparametric efficiency bound Veff=⟨PΓ,A−1PΓ⟩V_ eff= P_T ,A^-1P_T , but requires CAd¯/n→0C_A d/n→ 0 for the CLT. The whitened estimator achieves CLT under the weaker condition n≫d¯logc⁡d¯n d ^c d (no CAC_A), but its variance Vws≥VeffV_ ws≥ V_ eff may be larger. The gap vanishes when the Fisher information is nearly constant across comparisons: if ‖T⋆‖∞\|T \|_∞ is small, all matchups are competitive, I(η⋆)≈1/4I(η )≈ 1/4, and Vws≈VeffV_ ws≈ V_ eff. 5.2 Extension to non-uniform sampling via inverse-probability weighting In practice, pairwise comparison data are far from uniformly collected. As illustrated in Figure 1, popular frontier models receive disproportionately many battles, while prompts concentrate on engaging topics. This non-uniformity breaks the isotropic Gram structure on which the whitened estimator relies. IPW framework. Let p(x):=Π⋆(X=x)p(x):= (X=x) be the true sampling probability and q(x)=1/d⋆q(x)=1/d the uniform reference distribution. Define the importance weight w(x):=q(x)/p(x)w(x):=q(x)/p(x) and the IPW-whitened score s~ηw(y,η;x):=w(x)s~η(y,η). s_η^\,w(y,η;x):=w(x)\, s_η(y,η). (5.10) The weighted Gram operator ⟨H1,Gq(H2)⟩:=X∼p[w(X)⟨H1,X⟩⟨H2,X⟩]=X∼q[⟨H1,X⟩⟨H2,X⟩] H_1,G_q(H_2) :=E_X p\! [w(X)\, H_1,X H_2,X ]=E_X q\! [ H_1,X H_2,X ] (5.11) is exactly the uniform-design operator. On the column-sum-zero subspace, Gq(H)=1d⋆H,Aq−1=d⋆P.G_q(H)= 1d H, A_q^-1=d P_T. (5.12) Hence the oracle and plug-in directions are Hq,0=d⋆PΓ,H^q,0=d⋆P^Γ,H_q,0=d P_T , H_q,0=d P_T , (5.13) which are identical to the whitened directions (5.3). No operator inversion is needed. IPW one-step estimator and CLT. When the sampling law p(x)p(x) is known, the IPW-whitened one-step estimator is ψ^ipw=⟨Γ,T^⟩+ℙn[w(X)s~η(Y,η^)⟨H^q,0,X⟩]. ψ_ ipw= , T +P_n\! [w(X)\, s_η\! (Y, η )\, H_q,0,X ]. (5.14) Since the weighted Gram GqG_q is isotropic, the analysis of Theorems 5.1–5.2 carries through with each remainder term acquiring at most a multiplicative factor of ‖w‖∞≤Cp/cp\|w\|_∞≤ C_p/c_p. Theorem 5.3 (CLT for the IPW-whitened estimator). Under the conditions of Theorem 5.1 and the overlap condition cp/d⋆≤p(x)≤Cp/d⋆,c_p/d ≤ p(x)≤ C_p/d , (5.15) n(ψ^ipw−ψ(T⋆))Vws→(0,1), n ( ψ_ ipw-ψ(T ) ) V_ ws\; d\;N(0,1), where VwsV_ ws is defined in (5.8). The remainder bound is (5.6) with C(μ,κ,r,m)C(μ,κ,r,m) replaced by C(μ,κ,r,m,Cp/cp)C(μ,κ,r,m,C_p/c_p). The efficiency perspective. Alternatively, in the large-sample regime n≫CA2d¯logc⁡d¯n C_A^2 d ^c d, one could use the general one-step estimator of Section 4 with the original non-uniform operator G and achieve the semiparametric efficiency bound VeffV_ eff (which now depends on the sampling probabilities p(x)p(x)). The IPW approach is useful because it restores isotropy directly, eliminating the CAC_A bottleneck at the cost of replacing VeffV_ eff by Vws≥VeffV_ ws≥ V_ eff. The proof for the efficiency case follows from the same operator inversion argument as in Section 4, with G modified to incorporate the non-uniform design weights; see Appendix E. Unknown sampling distribution. When p(x)p(x) is unknown, it must be estimated from data. Write w^(x):=q(x)/p^(x) w(x):=q(x)/ p(x) for the feasible weight and ψ^ipw=⟨Γ,T^⟩+ℙn[w^(X)s~η(Y,η^)⟨H^q,0,X⟩]. ψ_ ipw= , T +P_n\! [ w(X)\, s_η\! (Y, η )\, H_q,0,X ]. (5.16) The key requirement is an entrywise relative error guarantee for p p: supx|p^(x)p⋆(x)−1|≤ϵp≪ 1 _x | p(x)p (x)-1 |\;≤\; _p\; \;1 (5.17) with high probability. This guarantee is achievable when the sampling mechanism has low effective dimension. For instance, in the matrix case (m=2m=2), if tasks are sampled independently of model pairs, each task category t with probability π(t)π(t) and each unordered pair a,b\a,b\ with probability q(a)q(b)q(a)q(b), then the effective parameter dimension is dpar=O(d1+d2)d_ par=O(d_1+d_2), and ϵp=O(dpar/n) _p=O( d_ par/n) is achievable by standard spectral methods. Our numerical experiments in Section 6 confirm that such structured sampling distributions are well-estimated in practice. The weight-estimation error does not affect the first-order limit. Since ⋆[s~η(Y,η⋆)∣X]=0E [ s_η(Y,η ) X]=0, the leading weight error is centered out: ⋆[(w^−w)s~η(Y,η⋆)⟨Hq,0,X⟩]=0.E [( w-w) s_η(Y,η ) H_q,0,X ]=0. The surviving contribution is a second-order Gram mismatch of order ϵp⋅‖Δ‖∞ _p·\| \|_∞, which is negligible. The complete proof, including the overlap condition (5.15), the entrywise relative error guarantee (5.17), and explicit remainder bounds, is given in Appendix E. 5.3 Extension to nonlinear functionals Many inferential targets in LLM evaluation are nonlinear: the win probability (1.1), category-weighted leaderboard scores, or calibrated transformations of latent scores. The score-whitening framework extends naturally to such targets by replacing the fixed gradient Γ with the plug-in gradient Γ^ψ:=∇ψ(T^) _ψ:=∇ψ( T). Assumption 5.4 (Finite-entry nonlinear functional). There exists a fixed index set Sψ⊂[d1]×⋯×[dm]S_ψ⊂[d_1]×·s×[d_m] with |Sψ|=sψ=O(1)|S_ψ|=s_ψ=O(1) and a twice continuously differentiable function g:ℝsψ→ℝg:R^s_ψ such that ψ(T)=g(TSψ)ψ(T)=g(T_S_ψ) in a neighborhood of T⋆T . Moreover, ‖∇g‖2≤Cg\|∇ g\|_2≤ C_g and ‖∇2g‖op≤Lg\|∇^2g\|_ op≤ L_g throughout this neighborhood. This assumption covers all standard LLM evaluation targets: a head-to-head win probability ψ(T)=σ(Ta,u−Tb,u)ψ(T)=σ(T_a,u-T_b,u) has sψ=2s_ψ=2; a smoothed win rate averaged over a finite reference pool has sψs_ψ equal to the pool size times the number of categories. The finite-support condition ensures ‖Γψ‖1=O(1)\| _ψ\|_1=O(1) and ‖Γ^ψ−Γψ‖1≲‖Δ‖∞\| _ψ- _ψ\|_1 \| \|_∞. The score-whitened one-step estimator is ψ^nl=ψ(T^)+ℙn[s~η(Y,η^)⟨H^ψ,X⟩],H^ψ:=d⋆P^Γ^ψ. ψ_ nl=ψ( T)+P_n\! [ s_η(Y, η)\, H_ψ,X ], H_ψ:=d P_T _ψ. (5.18) Theorem 5.5 (CLT for nonlinear functionals). Under the conditions of Theorem 5.2 and Assumption 5.4, n(ψ^nl−ψ(T⋆))→(0,Vws(ψ)),Vws(ψ)=⋆[⟨d⋆PΓψ,X⟩2I(η⋆)]. n ( ψ_ nl-ψ(T ) )\; d\;N\! (0,\;V_ ws(ψ) ), V_ ws(ψ)\;=\;E \! [ d P_T _ψ,X ^2I(η ) ]. The proof is deferred in Appendix E.3. The key idea is to reduce the nonlinear problem to the linear one via a three-term decomposition: (i) the linearized estimator with the true gradient Γψ _ψ, which is exactly the linear problem of Theorem 5.1; (i) a second-order plug-in Taylor remainder Rplug,ψ=O(‖Δ‖∞2)R_plug,ψ=O(\| \|_∞^2) from the nonlinear functional itself; and (i) a gradient-perturbation correction from using Γ^ψ _ψ in place of Γψ _ψ, which carries an extra factor of ‖Δ‖∞\| \|_∞ and is therefore lower order. 6 Numerical experiments This section evaluates the main theoretical claims using both synthetic and real-data experiments. We first show that low-rank initialization is essential in sparse pairwise-comparison settings and substantially improves over a naive per-task BTL fit. We then validate asymptotic normality and variance calibration for both linear and nonlinear targets, and study the non-uniform-sampling extension. Finally, we evaluate the proposed methods on real Arena dataset. Additional implementation details and supplementary experiments on robustness evaluation are deferred to Appendix A. All synthetic experiments use d1=d2=d=200d_1=d_2=d=200, rank r=5r=5, and the standard logistic link σ(η)=(1+e−η)−1σ(η)=(1+e^-η)^-1. The ground-truth matrix T⋆∈ℝd×dT ^d× d is generated as T⋆=ΘA⊤T = A , where Θ,A∈ℝd×r ,A ^d× r have i.i.d. standard normal entries. We then center and rescale T⋆T so that its signal strength satisfies ‖T⋆‖∞=α\|T \|_∞=α with α=5α=5. Except in the non-uniform-sampling experiment of Section 6.3, each observation (ui,pi,qi,Yi)(u_i,p_i,q_i,Y_i) is generated as follows: a task category uiu_i is sampled uniformly, two distinct models pi≠qip_i≠ q_i are sampled uniformly, and the comparison outcome is then drawn according to Yi∼Bernoulli(σ(Tpi,ui⋆−Tqi,ui⋆)).Y_i \! (σ\! (T _p_i,u_i-T _q_i,u_i ) ). Throughout, we compare the semiparametrically efficient one-step estimator ψ^eff ψ_ eff from (4.4) with the score-whitened estimator ψ^ws ψ_ ws from (5.4). All point estimates use K-fold cross-fitting with K=6K=6. The plug-in standard error SE SE is computed from the full sample by fitting T T on all observations, constructing the corresponding influence-function direction, and estimating the asymptotic variance by the empirical second moment of the estimated influence function. 6.1 Initialization and comparison with the baseline We first examine the quality of the initial low-rank estimator. We parameterize the latent score matrix in low-rank form as T=UV⊤,U,V∈ℝd×r,T=UV ,U,V ^d× r, and optimize the BTL log-likelihood over (U,V)(U,V) under this factorization. Since the problem is nonconvex, we start the procedure by a spectral estimator constructed from the observed pairwise-comparison matrix. Then we run an alternating minimization (AltMin) procedure to update V and U. After each round, we project the fitted matrix back to rank r by truncated SVD and clip its entries to the interval [−α0,α0][- _0, _0], with α0=α+2 _0=α+2, to stabilize the iterates and prevent extreme fitted logits. In our implementation, three alternating-minimization rounds are sufficient to obtain an accurate estimator T T. We then apply an entrywise refinement step to improve the accuracy of individual entries, which is the scale most relevant for the one-step inference procedures. Specifically, let T^=U^Σ^V^⊤ T= U V be the rank-r SVD of the alternating-minimization estimator, and write A^=V^Σ^1/2 A= V ^1/2 for the estimated column factor. Treating A A as fixed, we re-estimate each row loading of T⋆T by fitting a row-wise logistic regression using only the comparisons involving that row. This yields a refined estimate of the row factor. We then reverse the roles of rows and columns and similarly refine the column factor. The resulting estimator, denoted by T^ref T_ ref, preserves the global low-rank structure learned by alternating minimization while correcting local entrywise bias, and in our experiments it substantially improves estimation accuracy. Figure 2 reports the estimation error as a function of sample size, comparing alternating minimization alone (AltMin) with alternating minimization followed by refinement (AltMin + Refinement). Refinement yields clear improvements in both relative Frobenius error and entrywise ℓ∞ _∞ error once the sample size is moderately large. Moreover, to illustrate the value of the low-rank structure, we also compare with a naive approach that fits a separate BTL model independently within each task category, ignoring any shared structure across tasks. At n=60,000n=60,000, each task receives only about 300 observations spread across (d2)=19,900 d2=19,900 possible model pairs, so most pairs are never observed within a task. As shown in Table 2, the naive estimator performs dramatically worse: both its Frobenius and entrywise errors are an order of magnitude larger than those of the low-rank estimator (AltMin). This confirms that the low-rank assumption is not merely a modeling convenience, but is essential for stable estimation and inference in the sparse regime. Figure 2: Estimation error versus sample size: relative Frobenius error (left) and entrywise ℓ∞ _∞ error (right) for alternating minimization alone and alternating minimization followed by entrywise refinement. Refinement substantially improves the accuracy of the initial estimator. Table 2: Comparison of the naive per-task BTL estimator and the low-rank alternating-minimization estimator. Method ‖T^−T⋆‖F/‖T⋆‖F\| T-T \|_F/\|T \|_F ‖T^−T⋆‖∞\| T-T \|_∞ Mean |T^ij−Tij⋆|| T_ij-T _ij| Naive per-task BTL 4.01 12.3 2.69 AltMin 0.41 1.8 0.28 6.2 Asymptotic normality and variance calibration We evaluate the proposed inference procedures on two representative targets. The first is a linear target, ψ1(T⋆)=Ta0,u0⋆, _1(T )=T _a_0,u_0, which corresponds to the latent ability of model a0a_0 on task u0u_0. The second is a nonlinear target, ψ2(T⋆)=σ(Ta0,u0⋆−Tb0,u0⋆), _2(T )=σ\! (T _a_0,u_0-T _b_0,u_0 ), which represents the probability that model a0a_0 beats model b0b_0 on task u0u_0. Throughout the experiments, we fix (a0,u0,b0)=(1,1,2)(a_0,u_0,b_0)=(1,1,2). For each configuration, we run 500 independent replications. In each replication, we generate fresh data, construct the cross-fitted initial estimator (AltMin), compute both the efficient and score-whitened one-step estimators and their standard errors. Table 3 summarizes coverage and variance calibration. For the linear entry target, both estimators achieve near-nominal coverage and good variance calibration, confirming the asymptotic normality shown in Theorems 4.4 and 5.2. For the nonlinear win-probability target, the same qualitative pattern holds, although the finite-sample approximation is more demanding: coverage improves as n increases, consistent with the additional higher-order remainder in the nonlinear expansion of Section 5.3. Across all settings, the efficient estimator consistently yields smaller standard errors than the score-whitened estimator. This is consistent with the theory: the efficient estimator targets the semiparametric variance bound, while score whitening trades some efficiency for robustness by removing the CAC_A bottleneck. Our experiments favor the efficient estimator because they are conducted in regimes where the efficient direction can still be estimated stably. Table 3: Coverage and variance calibration for the entry target and the nonlinear win-probability target. SE^/SE⋆ SE/ SE is the ratio of the median plug-in standard error to the oracle asymptotic standard error computed from T⋆T . Target Method n Coverage med SE SE SE^/SE⋆ SE/ SE Entry Efficient 60K 0.942 0.377 1.029 Whitened 60K 0.940 0.481 1.041 Win-prob Efficient 60K 0.890 0.121 0.970 Whitened 60K 0.914 0.154 1.029 Efficient 80K 0.900 0.106 0.987 Whitened 80K 0.928 0.133 1.027 6.3 Non-uniform sampling We now consider non-uniform sampling, where the task category and model pair are drawn from heterogeneous distributions. Specifically, the task is sampled as u∼πJu _J and the two models are sampled independently as p,q∼πMp,q _M (resampling if p=qp=q), where πJ _J and πM _M are drawn from Dirichlet(5⋅d)Dirichlet(5·1_d). This yields mildly non-uniform sampling distributions, with task and model probabilities varying by roughly a factor of 10 between the most and least likely categories, which mimics the real Arena evaluation case. We compare four methods: the score-whitened estimator with IPW using known π, the same estimator using estimated π π, the efficient one-step estimator using the non-uniform information operator with known π, and its corresponding version using estimated π π. Table 4 reports the results. Three conclusions are clear. First, all four methods attain coverage close to the nominal 95% level, providing empirical support for the asymptotic normal approximation under non-uniform sampling. Second, using estimated π π instead of the true ones makes essentially no difference, which is consistent with the theory in Section 5.2: the first-order effect of weight estimation vanishes by conditional centering. Third, the efficient estimator retains a substantial variance advantage over the IPW-whitened estimator. This gap is even larger than under uniform sampling, because IPW interacts with score whitening and amplifies heavy-tailed contributions from under-sampled comparison types. Table 4: Results under non-uniform sampling for the linear entry target. Method Coverage med SE SE SE^/SE⋆ SE/ SE Whitened + IPW (known π) 0.942 0.574 1.063 Whitened + IPW (est. π π) 0.944 0.571 1.059 Efficient (known π) 0.936 0.372 1.044 Efficient (est. π π) 0.936 0.372 1.044 6.4 Real-data experiment on Arena We finally validate the proposed procedures on real pairwise-comparison data from Arena (Chiang et al., 2024), a platform that collects anonymous side-by-side evaluations of large language models from human users. We use the publicly available arena-human-preference-140k dataset (LMSYS, 2025). After filtering to the 30 most frequently appearing models and assigning each conversation to one of 10 non-overlapping task categories, such as math, coding, and creative writing, based on metadata tags, we obtain N=81,150N=81,150 observations organized as a 30×1030× 10 model-by-category comparison matrix. Here Tij⋆T _ij denotes the latent BTL ability parameter of model i in task category j. The category distribution is highly non-uniform: the largest category contains about 7.17.1 times as many observations as the smallest category. This makes the dataset a natural test bed for the non-uniform-sampling setting studied in Section 5.2. Further details are in Appendix A.4. To assess the low-rank assumption in the real-data setting, we first construct a full-data benchmark matrix by fitting separate BTL models within each task category via Newton’s method. The singular values of this benchmark matrix decay rapidly, and the first three singular components explain about 95% of its total squared Frobenius norm. We therefore use rank r=3r=3 in the real-data experiments. This suggests that the dominant cross-category variation is well captured by a low-rank structure, while still allowing some model misspecification. We subsample 20% of the data for training and compare three methods: the naive per-category BTL baseline, the score-whitened estimator with IPW correction, and the efficient one-step estimator. In the main paper, we report only two representative targets: (1) the linear entry target T⋆[Gemini-2.5-Pro,math]T [Gemini-2.5-Pro,\,math] that measures the latent ability of Gemini-2.5-Pro on math task; and (2) the nonlinear win-probability target P(Gemini-2.5-Pro>Claude-Opus-4∣math)P(Gemini-2.5-Pro>Claude-Opus-4 ). Experiments on additional target functionals as well as the efficiency gain over varying subsample sizes are deferred to Appendix A.4. Table 5: Real-data inference on Arena for one linear target and one nonlinear target. Naive denotes separate per-category BTL modeling; Whitened denotes score-whitening with IPW; Efficient denotes the one-step estimator based on the efficient influence function. Target ψ⋆ψ Method Emp. std med SE SE Coverage T⋆[Gemini,math]T [Gemini,\,math] +0.687+0.687 Naive 0.199 0.274 0.998 Whitened 0.355 0.523 0.986 Efficient 0.151 0.163 0.930 P(Gemini>Claude∣math)P(Gemini>Claude ) +0.652+0.652 Naive 0.055 0.061 0.954 Whitened 0.115 0.170 0.986 Efficient 0.047 0.052 0.954 Table 5 summarizes the real-data results over 500 replications. Our efficient one-step estimator produces the narrowest intervals and coverage closest to the nominal 95% level, while the naive and score-whitened estimators are more conservative. Relative to the full-data per-category BTL benchmark, the naive estimator is correctly specified, whereas the low-rank estimators trade a small approximation error for substantial variance reduction by borrowing strength across categories. The score-whitened estimator further sacrifices efficiency for robustness, which leads to wider intervals. This phenomenon is consistent with the theory. References Bose et al. 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Appendix A provides extra simulation and real-data evidence omitted from the main text. Appendix B develops the semiparametric efficiency arguments and proofs for Section 3. Appendix C provides the detailed proofs for Section 4, beginning with the initial condition properties (Section C.1) and the proof strategy. Appendix D and Appendix E provide proofs for Section 5 that covers score whitening, inverse-probability weighting, and the nonlinear-functional extension. Appendix F provides the key Berry–Esseen bound for the pairwise-comparison estimators, used in the main proof. Appendix G includes broader implications of the inference framework for tensor completion. Appendix H provides a construction of the entrywise-accurate initial estimator required by Assumption 4.1. Appendix A Additional numerical experiments This appendix collects additional numerical results for Section 6. We provide CLT diagnostics for both linear and nonlinear targets, broader robustness checks across sample size and rank, further diagnostics under non-uniform sampling, and additional real-data analyses on Arena, including preprocessing details and sensitivity to the subsample-fractions. A.1 Additional CLT diagnostics Figure 3 displays the z-score histograms for the linear entry target with n=60,000n=60,000 (n/d2=1.5n/d^2=1.5). Both the one-step efficient and score-whitened estimators exhibit close agreement with the (0,1)N(0,1) reference density, confirming the CLT. Figure 3: CLT validation for the entry target (α=5α=5, n=60,000n=60,000, d=200d=200, K=6K=6): z-score histograms for the score-whitened (left) and efficient (right) estimators, compared to the (0,1)N(0,1) reference density. Both methods exhibit close agreement with the standard normal. Moreover, Figure 4 reports the histogram diagnostics for the nonlinear win-probability target in the synthetic experiment. The nonlinear target requires larger sample sizes for the Gaussian approximation to stabilize, which is consistent with the additional higher-order remainder terms in Section 5.3. Figure 4: CLT validation for the nonlinear win-probability target. Top row: n=60,000n=60,000. Bottom row: n=80,000n=80,000. Left: score-whitened estimator. Right: efficient estimator. A.2 Additional robustness checks We further examine robustness to the sample size and the rank, while fixing d=200d=200 and α=5α=5. Figure 5 plots the 95%95\% CI coverage and variance calibration for the entry target as n/d2n/d^2 increases from 0.750.75 to 2.52.5. Both estimators maintain coverage close to the nominal 95% level, with the efficient estimator showing consistently stable variance calibration. The score-whitened estimator is more variable at smaller sample sizes, but improves as n grows. Figure 6 reports the same quantities at fixed n=60,000n=60,000 as the rank varies over r∈2,3,5,8,10r∈\2,3,5,8,10\. Coverage remains satisfactory across all ranks for both methods. As the rank increases, the score-whitened estimator becomes more variable, whereas the efficient estimator remains well calibrated. Figure 5: Coverage (left) and variance calibration (right) for the entry target as a function of n/d2n/d^2 (d=200d=200, r=5r=5, α=5α=5, K=6K=6, 500 replications). Figure 6: Coverage (left) and variance calibration (right) for the entry target as a function of rank r (d=200d=200, α=5α=5, n=60,000n=60,000, K=6K=6, 500 replications). A.3 Additional diagnostics under non-uniform sampling Figure 7 reports the z-score histograms for the four methods in the non-uniform-sampling experiment. All four estimators exhibit approximately Gaussian behavior, and the known-π and estimated-π π versions are nearly indistinguishable. Figure 7: CLT diagnostics for the non-uniform-sampling experiment. A.4 Real-data preprocessing and robustness evaluation We use a public Arena snapshot containing about 140,000 pairwise comparisons. After filtering to the 30 most frequently appearing models and aggregating prompts into 10 task categories based on metadata tags, we obtain 81,150 observations. Table 6 lists the category counts. The distribution is substantially non-uniform: the largest category (code_technical, n=19,091n=19,091, 23.5%) contains 7.1×7.1× more observations than the smallest (code_general, n=2,699n=2,699, 3.3%). Table 6: Task categories and counts after filtering to the top 30 models in the Arena dataset. Category n % code_technical 19,091 23.5 general 14,682 18.1 creative_practical 12,395 15.3 math 6,583 8.1 creative_writing 6,432 7.9 instruction_following 5,885 7.3 analytical 5,686 7.0 creative_abstract 4,744 5.8 domain_knowledge 2,953 3.6 code_general 2,699 3.3 Total 81,150 100.0 Table 7 reports the full real-data results for all four inference targets. The efficient one-step estimator achieves coverage between 0.930 and 0.976, the closest to the nominal 95% level among the three methods, and produces the narrowest confidence intervals. Table 7: Full real-data results on Arena. Target ψ⋆ψ Method Emp. std med SE SE Coverage T⋆[Gemini,math]T [Gemini,\,math] +0.687+0.687 Naive 0.199 0.274 0.998 Whitened 0.355 0.523 0.986 Efficient 0.151 0.163 0.930 Gemini–Claude (math) +0.630+0.630 Naive 0.248 0.271 0.980 Whitened 0.503 0.736 0.984 Efficient 0.203 0.226 0.954 P(Gemini>Claude∣math)P(Gemini>Claude ) +0.652+0.652 Naive 0.055 0.061 0.954 Whitened 0.115 0.170 0.986 Efficient 0.047 0.052 0.954 Gemini–o3 (code_tech) +0.273+0.273 Naive 0.140 0.155 0.976 Whitened 0.378 0.476 0.976 Efficient 0.116 0.141 0.976 Finally, to examine how the benefit of the low-rank efficient estimator changes with data availability, we repeat the naive-versus-efficient comparison at subsample fractions of 10%10\%, 5%5\%, and 2%2\%; see Table 8. The advantage of the efficient estimator becomes more pronounced as the sample size decreases. At 20%20\% subsampling, both methods perform reasonably well, with the efficient estimator achieving about 1.24×1.24× smaller empirical standard deviation on average. As the fraction decreases to 10%10\% and 5%5\%, this gain increases to about 1.47×1.47× and 1.50×1.50×, respectively, while coverage remains adequate. At 2%2\% subsampling, the naive BTL estimator becomes highly unstable, whereas the efficient estimator remains usable, though with some mild undercoverage. Overall, these results show that the low-rank structure becomes increasingly valuable in data-scarce regimes, where borrowing strength across task categories leads to substantially tighter inference. Table 8: Naive versus efficient estimator at different subsample fractions of the Arena data. Naive BTL Efficient Frac nsubn_ sub Emp. std Cov Emp. std Cov Std ratio 20% 16,230 0.160 0.977 0.129 0.954 1.24× 10% 8,115 0.298 0.956 0.203 0.952 1.47× 5% 4,058 0.379 0.945 0.252 0.932 1.50× 2% 1,623 2.245 0.947 0.408 0.915 5.50× Appendix B Proofs for Section 3 B.1 Proof of Theorem 3.5 Fix any H∈H and consider the one-dimensional submodel Tε=T⋆+εHT_ =T + H (with the nuisance held fixed at Π⋆ ). By Assumption 3.4, Tε,Π[ψ^]=ψ(Tε)∀ε.E_T_ , [ ψ]=ψ(T_ ) ∀ . Differentiating both sides at ε=0 =0 and using (3.3) gives dεTε,Π[ψ^]|ε=0=DψT⋆(H). . dd E_T_ , [ ψ] |_ =0=D _T (H). On the other hand, writing (H)(X,Y):=sη(Y,η⋆)⟨H,X⟩S(H)(X,Y):=s_η(Y,η ) H,X for the directional score, the standard score identity yields dεTε,Π⋆[ψ^]|ε=0=⋆[ψ^⋅∑i=1nsη(Yi,ηi⋆)⟨H,Xi⟩]. . dd E_T_ , [ ψ] |_ =0=E \! [ ψ· _i=1^ns_η(Y_i, _i ) H,X_i ]. Using ⋆[ψ^]=ψ(T⋆)E [ ψ]=ψ(T ) and ⋆[sη(Y,η⋆)∣X]=0E [s_η(Y,η ) X]=0, the right-hand side equals ⋆[(ψ^−ψ(T⋆))⋅∑i=1nsη(Yi,ηi⋆)⟨H,Xi⟩].E \! [( ψ-ψ(T ))· _i=1^ns_η(Y_i, _i ) H,X_i ]. Applying Cauchy–Schwarz and i.i.d. sampling gives DψT⋆(H)2≤Var⋆(ψ^)⋅⋆[(∑i=1nsη(Yi,ηi⋆)⟨H,Xi⟩)2]=Var⋆(ψ^)⋅n⋆[sη(Y,η⋆)2⟨H,X⟩2],D _T (H)^2 ( ψ)·E \! [ ( _i=1^ns_η(Y_i, _i ) H,X_i )^2 ]=Var ( ψ)· n\,E [s_η(Y,η )^2 H,X ^2], which implies the directional bound. The sharp form (3.12) follows by maximizing the Rayleigh quotient ⟨P∇ψ(T⋆),H⟩2/⟨H,AH⟩ P_T∇ψ(T ),H ^2/ H,A\,H over H∈H , whose maximizer is H∝A−1P∇ψ(T⋆)H A^-1P_T∇ψ(T ) under Assumption 3.3. That is, the submodel in the direction H⋆H derived in Section 3.2 is the hardest to learn among all directions H∈H . Under Assumption 3.3, the information equation AH⋆=P∇ψ(T⋆)A\,H =P_T∇ψ(T ) has the unique solution H⋆=A−1P∇ψ(T⋆)H =A^-1P_T∇ψ(T ). The efficient influence function is ϕEIF(X,Y)=sη(Y,η⋆)⟨A−1P∇ψ(T⋆),X⟩, _EIF(X,Y)=s_η(Y,η ) A^-1P_T∇ψ(T ),\,X , and satisfies ⋆[ϕEIF2]=Veff(ψ)E [ _EIF^2]=V_eff(ψ), showing that the lower bound (3.12) is attainable. ∎ B.2 Semiparametric derivation of the EIF We provide the full semiparametric argument establishing that ϕEIF _EIF defined in (3.10) is the efficient influence function. Recall the joint density factorization pT,Π(X,Y)=gΠ(X)p(Y∣⟨T,X⟩)p_T, (X,Y)=g_ (X)\,p(Y T,X ). Define the signal tangent space in L2(P⋆)L^2(P ) by T:=sη(Y,η⋆)⟨H,X⟩:H∈¯L2(P⋆)T_T:= \s_η(Y,η ) H,X :H \^L^2(P ), and the nuisance tangent space X:=h(X):⋆[h(X)]=0,⋆[h(X)2]<∞T_X:=\h(X):E [h(X)]=0,\ E [h(X)^2]<∞\. Since ⋆[sη(Y,η⋆)∣X]=0E [s_η(Y,η ) X]=0, it follows that T⟂XT_T _X in L2(P⋆)L^2(P ), and the full tangent space is the orthogonal direct sum =T⊕XT=T_T _X. An influence function (IF) for a regular estimator of ψ(T⋆)ψ(T ) is any ϕ∈L02(P⋆)φ∈ L^2_0(P ) satisfying DψT⋆(H)=⋆[ϕ(X,Y)ℓ˙ε(X,Y)]D _T (H)=E \! [φ(X,Y)\, _ (X,Y) ] for every regular submodel ε↦(Tε,Πε) (T_ , _ ), where ℓ˙ε _ is the full model score. Since ψ depends only on T, the derivative vanishes for pure Π -perturbations, and the identity constrains ϕφ only through its projection onto TT_T. It follows that IFs are not unique (ϕ+uφ+u is an IF whenever u∈⟂u ), and the minimum-variance IF is ϕEIF:=Proj(ϕ0)∈T _EIF:=Proj_T( _0) _T for any IF ϕ0 _0. Since ϕEIF∈T _EIF _T, there exists H⋆∈H such that ϕEIF(X,Y)=sη(Y,η⋆)⟨H⋆,X⟩ _EIF(X,Y)=s_η(Y,η ) H ,X . The score identity for all signal submodels Tε=T⋆+εHT_ =T + H yields ⟨P∇ψ(T⋆),H⟩=⋆[sη(Y,η⋆)2⟨H⋆,X⟩⟨H,X⟩]=⟨H⋆,AH⟩,∀H∈, P_T∇ψ(T ),H =E \! [s_η(Y,η )^2 H ,X H,X ]= H ,A\,H , ∀ H , where A=PGPA=P_TGP_T, giving the information equation AH⋆=P∇ψ(T⋆)AH =P_T∇ψ(T ). B.3 Detailed computations for the pairwise comparison model We present the detailed computation of the information operator and the efficiency bound for the pairwise comparison model discussed in Section 3. Example B.1. We consider the pairwise comparison model where the true signal matrix belongs to r,0:=T∈ℝd1×d2:rank(T)=r, 1d1⊤T=0.P_r,0:= \T ^d_1× d_2:\ rank(T)=r,\ \ 1_d_1 T=0 \. Here d1d_1 indexes the items (first mode) and d2d_2 indexes the arenas/users (second mode). The first-mode sum-to-zero constraint selects a unique representative from each shift-equivalence class and removes the unidentifiable directions (shifts along the item mode) in preference-based observations; see also Example 3.1. At each round we draw an arena R∼Unif([d2])R ([d_2]) and an unordered pair of items (J,K)(J,K) uniformly from (j,k):1≤j<k≤d1\(j,k):1≤ j<k≤ d_1\, and observe Y∣(R,J,K)∼Bernoulli(g(ηJK,R⋆)),ηjk,r⋆:=Tj,r⋆−Tk,r⋆,Y (R,J,K) (g(η _JK,R) ), η _jk,r:=T _j,r-T _k,r, where g(z)=1/(1+e−z)g(z)=1/(1+e^-z) and I(z):=g(z)(1−g(z))I(z):=g(z)(1-g(z)). Define the design matrix X:=(eJ−eK)eR⊤∈ℝd1×d2,⟨H,X⟩=HJ,R−HK,R.X:=(e_J-e_K)e_R ^d_1× d_2, H,X =H_J,R-H_K,R. The score along any parametric submodel Tε=T⋆+εHT_ =T + H is S(H)(Z)=(Y−g(η⋆))⟨H,X⟩=(Y−g(η⋆))(HJ,R−HK,R),Z=(R,J,K,Y).S(H)(Z)=(Y-g(η ))\, H,X =(Y-g(η ))\,(H_J,R-H_K,R), Z=(R,J,K,Y). Recall that the tangent space is (cf. Example 3.1) r,0=UA⊤+QCV⊤:A∈ℝd2×r,C∈ℝ(d1−1)×r,T_r,0=\\,UA +QCV :\ A ^d_2× r,\ C ^(d_1-1)× r\,\, where Q∈ℝd1×(d1−1)Q ^d_1×(d_1-1) has orthonormal columns spanning d1⟂1_d_1 . For any direction H∈r,0H _r,0, the Fisher quadratic form is ‖S(H)‖L2(P⋆)2=[I(η⋆)⟨H,X⟩2]=[I(η⋆)(HJ,R−HK,R)2].\|S(H)\|_L^2(P )^2=E [I(η ) H,X ^2 ]=E [I(η )(H_J,R-H_K,R)^2 ]. Conditioning on R=rR=r and writing hr∈ℝd1h_r ^d_1 for the r-th column of H, define the (weighted) Laplacian Lr:=1(d12)∑1≤j<k≤d1wr,jk(ej−ek)(ej−ek)⊤,wr,jk:=I(Tj,r⋆−Tk,r⋆)∈(0,1/4].L_r:= 1 d_12 _1≤ j<k≤ d_1w_r,jk\,(e_j-e_k)(e_j-e_k) , w_r,jk:=I(T _j,r-T _k,r)∈(0,1/4]. Then ‖S(H)‖L2(P⋆)2=1d2∑r=1d2hr⊤Lrhr.\|S(H)\|_L^2(P )^2= 1d_2 _r=1^d_2h_r L_rh_r. Moreover, for each r and any u∈ℝd1u ^d_1, u⊤Lru=1(d12)∑j<kwr,jk(uj−uk)2,u L_ru= 1 d_12 _j<kw_r,jk(u_j-u_k)^2, so ker⁡(Lr)=span⁡d1 (L_r)=span\1_d_1\ and Lr≻0L_r 0 on d1⟂1_d_1 . Since H∈r,0H _r,0 implies d1⊤hr=01_d_1 h_r=0 for all r, we get ‖S(H)‖L2(P⋆)2>0for all H∈r,0∖0.\|S(H)\|_L^2(P )^2>0 all H _r,0 \0\. Hence the restricted Fisher information operator A:=P∘G∘P:r,0→r,0A:=P_T G P_T:\ T_r,0 _r,0 is injective (and, since r,0T_r,0 is finite-dimensional, invertible). This is precisely the formal sense in which the first-mode sum-to-zero normalization removes the singular kernel present in pairwise models without normalization (shifts along the item mode). Let ψ(T)=⟨Γ,T⟩ψ(T)= ,T be any linear functional. Its pathwise derivative along H is DψT⋆(H)=⟨Γ,H⟩D _T (H)= ,H . Let b:=P(Γ)∈r,0b:=P_T( ) _r,0 be the orthogonal projection of Γ onto r,0T_r,0. Then the information inequality yields VarT⋆(ψ^n)≥1nsupH∈r,0∖0⟨b,H⟩2⟨H,AH⟩=1n⟨b,A−1b⟩.Var_T ( ψ_n)\ ≥\ 1n _H _r,0 \0\ b,H ^2 H,A\,H \ =\ 1n\, b,A^-1b . The supremum is attained at the unique hardest direction H⋆=A−1b∈r,0,i.e.AH⋆=b.H =A^-1b _r,0, .e. A\,H =b. The corresponding efficient influence function is ϕEIF(Z)=S(H⋆)(Z)=(Y−g(η⋆))(HJ,R⋆−HK,R⋆), _EIF(Z)=S(H )(Z)=(Y-g(η ))\,(H _J,R-H _K,R), and it satisfies [ϕEIF]=0,[ϕEIF2]=⟨b,A−1b⟩,E[ _EIF]=0, [ _EIF^2]= b,A^-1b , so the lower bound is sharp and is achieved by the (oracle) one-step estimator ψ~n:=ψ(T⋆)+1n∑t=1nϕEIF(Zt). ψ_n:=ψ(T )+ 1n _t=1^n _EIF(Z_t). We can also provide an explicit matrix form characterization. Recall that every direction H in tangent space can be indexed by (A,C)(A,C) via the linear map H(A,C):=UA⊤+QCV⊤,A∈ℝd2×r,C∈ℝ(d1−1)×r.H(A,C)\;:=\;UA \;+\;QCV , A ^d_2× r,\;C ^(d_1-1)× r. We first give the Fisher information bilinear form. In the pairwise-logistic model, for each arena r∈[d2]r∈[d_2] there exists a symmetric PSD matrix Lr∈ℝd1×d1L_r ^d_1× d_1 (a weighted Laplacian) such that for any two tangent directions H1,H2H_1,H_2, T⋆[S(H1)S(H2)]=⟨H1,G(H2)⟩=∑r=1d2h1,r⊤Lrh2,r,E_T [S(H_1)S(H_2)]= H_1,G(H_2) = _r=1^d_2h_1,r L_r\,h_2,r, where hm,r∈ℝd1h_m,r ^d_1 is the r-th column of HmH_m. In the uniform pair sampling case, Lr=1(d12)∑1≤j<k≤d1wr,jk(ej−ek)(ej−ek)⊤,wr,jk:=I(Tj,r⋆−Tk,r⋆)=g(ηjk,r⋆)(1−g(ηjk,r⋆)).L_r\;=\; 1 d_12 _1≤ j<k≤ d_1w_r,jk\,(e_j-e_k)(e_j-e_k) , w_r,jk:=I(T _j,r-T _k,r)=g(η _jk,r)(1-g(η _jk,r)). Define the block diagonal matrix L:=blkdiag(L1,…,Ld2)∈ℝ(d1d2)×(d1d2).L:=blkdiag(L_1,…,L_d_2) ^(d_1d_2)×(d_1d_2). Now define the induced bilinear form on the coordinate space (A,C)(A,C) by ℬ((A1,C1),(A2,C2)):=∑r=1d2hr(A1,C1)⊤Lrhr(A2,C2),hr(A,C):=r-th column of H(A,C).B ((A_1,C_1),(A_2,C_2) ):= _r=1^d_2h_r(A_1,C_1) L_r\,h_r(A_2,C_2), h_r(A,C):=$r$-th column of H(A,C). Equip the coordinate space with the Euclidean inner product ⟨(A1,C1),(A2,C2)⟩θ:=⟨A1,A2⟩+⟨C1,C2⟩ (A_1,C_1),(A_2,C_2) _θ:= A_1,A_2 + C_1,C_2 . We use the standard identities: vec(AXB)=(B⊤⊗A)vec(X),vec(X⊤)=m,nvec(X)(X∈ℝm×n),vec(AXB)=(B A)vec(X), (X )=K_m,nvec(X)\ \ (X ^m× n), where ⊗ is the Kronecker product and m,nK_m,n is the commutation matrix. Then we derive the Jacobian J of the linear map (A,C)↦H(A,C)(A,C) H(A,C) in vectorized coordinates. Let θ:=[vec(A)vec(C)]∈ℝd2r+(d1−1)r,vec(H(A,C))∈ℝd1d2.θ:= bmatrixvec(A)\\ vec(C) bmatrix ^d_2r+(d_1-1)r, (H(A,C)) ^d_1d_2. Because H(A,C)H(A,C) is linear in (A,C)(A,C), there exists a (constant) matrix J such that vec(H(A,C))=Jθ.vec(H(A,C))=J\,θ. We compute J explicitly. vec(UA⊤)=(Id2⊗U)vec(A⊤)=(Id2⊗U)d2,rvec(A).vec(UA )=(I_d_2 U)vec(A )=(I_d_2 U)\,K_d_2,r\,vec(A). Hence JA:=(Id2⊗U)d2,r∈ℝ(d1d2)×(d2r).J_A:=(I_d_2 U)\,K_d_2,r ^(d_1d_2)×(d_2r). Similarly, vec(QCV⊤)=(V⊗Q)vec(C),vec(QCV )=(V Q)vec(C), so JC:=(V⊗Q)∈ℝ(d1d2)×((d1−1)r).J_C:=(V Q) ^(d_1d_2)×((d_1-1)r). Combining the two Jacobians, we have J=[JAJC].J= bmatrixJ_A&J_C bmatrix. Using the above representation on (A,C)(A,C), and ℬ((A1,C1),(A2,C2))=vec(H1)⊤Lvec(H2)B((A_1,C_1),(A_2,C_2))=vec(H_1) L\,vec(H_2), we obtain ℬ((A1,C1),(A2,C2))=θ1⊤(J⊤LJ)⏟=⁣:Kθ2.B ((A_1,C_1),(A_2,C_2) )= _1 (J LJ)_=:K\, _2. Therefore, the unique linear operator K satisfying ℬ(θ1,θ2)=⟨θ1,Kθ2⟩B( _1, _2)= _1,K _2 (Riesz representation under the Euclidean inner product on θ) has the matrix form K=J⊤blkdiag(L1,…,Ld2)J.K=J \,blkdiag(L_1,…,L_d_2)\,J. (B.1) We also present the matrix K in block forms where KAA K_A =JA⊤LJA=d2,r⊤(Id2⊗U⊤)L(Id2⊗U)d2,r, =J_A LJ_A=K_d_2,r (I_d_2 U )\,L\,(I_d_2 U)\,K_d_2,r, KAC K_AC =JA⊤LJC=d2,r⊤(Id2⊗U⊤)L(V⊗Q), =J_A LJ_C=K_d_2,r (I_d_2 U )\,L\,(V Q), KCC K_C =JC⊤LJC=(V⊤⊗Q⊤)L(V⊗Q), =J_C LJ_C=(V Q )\,L\,(V Q), KCA K_CA =KAC⊤. =K_AC . We can also calculate the gradient of a linear functional in (A,C)(A,C)-coordinates. For ψ(T)=⟨Γ,T⟩ψ(T)= ,T , the directional derivative along H(A,C)H(A,C) is DψT⋆[H(A,C)]=⟨Γ,UA⊤⟩+⟨Γ,QCV⊤⟩=⟨Γ⊤U,A⟩+⟨Q⊤ΓV,C⟩.D _T [H(A,C)]= ,UA + ,QCV = U,\ A + Q V,\ C . Hence the coordinate gradient of DψT⋆D _T is ∇Aψ=Γ⊤U∈ℝd2×r,∇Cψ=Q⊤ΓV∈ℝ(d1−1)×r. _Aψ= U ^d_2× r, _Cψ=Q V ^(d_1-1)× r. (B.2) Equivalently, in vectorized coordinates gθ:=[vec(∇Aψ)vec(∇Cψ)]=[vec(Γ⊤U)vec(Q⊤ΓV)].g_θ:= bmatrixvec( _Aψ)\\ vec( _Cψ) bmatrix= bmatrixvec( U)\\ vec(Q V) bmatrix. Finally we can solve the information equation, EIF, and efficiency bound in a matrix form. Let θ⋆=(A⋆,C⋆)θ =(A ,C ) denote a solution of the coordinate-space information equation Kθ⋆=gθ,K\,θ =g_θ, (B.3) where one can take θ⋆=K†gθ =K g_θ using the Moore–Penrose pseudoinverse, as we have proved that in the tangent space, the solution to information equation is unique. Define H⋆:=H(A⋆,C⋆)=UA⋆⊤+QC⋆V⊤H :=H(A ,C )=UA +QC V . Then the efficient influence function is ϕEIF(R,J,K,Y)=(Y−g(ηJK,R⋆))(HJ,R⋆−HK,R⋆), _EIF(R,J,K,Y)=(Y-g(η _JK,R))\,(H _J,R-H _K,R), (B.4) and the corresponding semiparametric efficiency bound takes the matrix form VarT⋆(ψ^n)≥1ngθ⊤K†gθ.Var_T ( ψ_n)\ ≥\ 1n\,g_θ K g_θ. (B.5) Appendix C Detailed Proofs for Section 4 C.1 Properties of the Initial Estimator (Initial Conditions) In the pairwise-comparison (and 1-bit) setting with bounded signal ‖T⋆‖∞≤B\|T \|_∞≤ B, the noise level is of constant order: the Fisher information satisfies cB≤I(η)≤1/4c_B≤ I(η)≤ 1/4, so there is no explicit noise parameter σ (equivalently, σ=O(1)σ=O(1)). Moreover, by incoherence and the Tucker decomposition, ‖T⋆‖F≍d⋆\|T \|_F d (since entries are O(B)O(B) and spread by incoherence), so by (2.10), λmin≍d⋆ _ d when κ and r are bounded. In particular, the signal-to-noise ratio satisfies λmin/σ≍d⋆→∞ _ /σ d →∞, and the SNR condition is automatically satisfied whenever n≫d¯logc⁡d¯n d ^c d. By Ma and Xia (2024), under Assumption 4.1 (entrywise accuracy of T T) together with the bounded signal condition (2.11) and the sample size condition (4.7), the subspace perturbation properties needed in the proof of Theorems 4.4 and 5.2 follow automatically from standard perturbation theory for low-rank tensors. We record these properties here. Proposition C.1 (Subspace perturbation from entrywise accuracy). Under Assumption 4.1, the bounded signal condition (2.11), and the sample size condition (4.7), the following hold for each mode j∈[m]j∈[m] with probability at least 1−n−c1-n^-c: (P3) Subspace perturbation control. Let U^j U_j denote the estimated singular subspace along mode j. Then ‖PU^j−PUj‖2 \|P_ U_j-P_U_j\|_2 ≤C(μ,κ,r,m)σλmind⋆d¯logc⁡d¯n, ≤ C(μ,κ,r,m) σ _min d d ^c dn, (C.1) ‖PU^j−PUj‖2,∞ \|P_ U_j-P_U_j\|_2,∞ ≤C(μ,κ,r,m)σλminμrjdjd⋆d¯logc⁡d¯n, ≤ C(μ,κ,r,m) σ _min μ r_jd_j d d ^c dn, (C.2) ‖PU^j−PUj‖∞ \|P_ U_j-P_U_j\|_∞ ≤C(μ,κ,r,m)σλminμrjdjd⋆d¯logc⁡d¯n. ≤ C(μ,κ,r,m) σ _min μ r_jd_j d d ^c dn. (C.3) (P4) Estimated incoherence. For all a∈[dj]a∈[d_j], ‖ea⊤U^j‖22≤2μrj/dj.\|e_a U_j\|_2^2≤ 2μ r_j/d_j. (P5) Projection stability. For any tensor A, ‖^()‖∞≤(∏j=1m2μrj)‖∞,\| P(A)\|_∞≤ ( _j=1^m 2μ r_j )\|A\|_∞, where ^():=×1PU^1⋯×mPU^m. P(A):=A×_1P_ U_1·s×_mP_ U_m. Proof. The spectral-norm subspace bound (P3, first line) follows from the Davis–Kahan sin⁡Θ theorem applied to the mode-j unfolding of T T versus T⋆T , using the entrywise accuracy to control the perturbation in spectral norm. The row-wise (ℓ2,∞ _2,∞) and entrywise (ℓ∞ _∞) bounds follow from standard leave-one-out or row-wise refinement arguments as in Ma and Xia (2024). Properties (P4) and (P5) are immediate consequences of (P3) under the incoherence condition (2.6) and the triangle inequality, once the subspace perturbation is sufficiently small. We omit the detailed verification, which follows standard arguments; see, e.g., Ma and Xia (2024, Section 3) for the general tensor case. ∎ Remark C.2 (Simplification for pairwise comparisons and 1-bit observations). Throughout the proofs in this appendix and Appendices D–E, we state all bounds in terms of the general noise parameter σ and λmin _ to maintain consistency with the additive-noise tensor completion setting (see Appendix G). In the pairwise-comparison (and 1-bit) case, σ=O(1)σ=O(1) and λmin≍d⋆ _ d , so every factor of σ/λminσ/ _ and d⋆ d cancels, and all error bounds reduce to d¯logc⁡d¯/n d ^c d/n plus lower-order log and constant terms. C.2 Setup and Notation We consider i.i.d. samples (Yi,Xi)i=1n\(Y_i,X_i)\_i=1^n from a pairwise comparison design with a non-constant Fisher information weight. C.2.1 Comparison sampling model Fix one “comparison mode” of size d1d_1 and denote by D:=∏k=2mdkD:= _k=2^md_k the number of contexts (the product of the remaining mode sizes). A single design draw X is generated as: (i) sample a context u∈[D]u∈[D] uniformly; (i) sample an unordered pair a,b⊂[d1]\a,b\⊂[d_1] uniformly among (d12) d_12 pairs; (i) set X=eu,a−eu,bX=e_u,a-e_u,b (a signed two-sparse tensor after flattening the non-comparison modes into u). Thus ‖X‖F2=2 X _F^2=2 and ⟨H,X⟩=Hu,a−Hu,b. H,X =H_u,a-H_u,b. (C.4) C.2.2 Column-sum-zero restriction We assume column-sum-zero along the comparison mode: for any H in the target subspace (in particular H⋆,H^H , H and their difference), ∑a=1d1Ha,u=0,∀u∈[D]. _a=1^d_1H_a,u=0, ∀ u∈[D]. (C.5) C.2.3 Effective dimension d⋆d We define d⋆:=∏j=1mdj,d¯:=maxj∈[m]⁡dj.d := _j=1^md_j, d:= _j∈[m]d_j. (C.6) Under the pairwise comparison model on mode 1, the effective sample-space size is D(d1−1)/2=d⋆⋅(d1−1)/(2d1)≍d⋆/2D(d_1-1)/2=d ·(d_1-1)/(2d_1) d /2. Since they differ by at most a constant factor, all bounds in this document hold up to absolute constants under either convention. C.2.4 Score and Fisher information Let η⋆=⟨T⋆,X⟩η = T ,X be the true linear predictor. We assume: Assumption C.3 (Score normalization and Fisher comparability). There exist constants σ>0σ>0 and C0≥1C_0≥ 1 such that for all x, ⋆[sη(Y,η⋆(x))∣X=x]=0,⋆[sη(Y,η⋆(x))2I(η⋆(x))|X=x]≤C02,E \! [s_η(Y,η (x)) X=x ]=0, \! [ s_η(Y,η (x))^2I(η (x))\ |\ X=x ]≤ C_0^2, (C.7) and the Fisher information satisfies c−σ2≤I(η⋆(x))≤c+σ2for some constants 0<c−≤c+<∞. c_-σ^2\ ≤\ I(η (x))\ ≤\ c_+σ^2 some constants 0<c_-≤ c_+<∞. (C.8) We further assume: (i) (Score derivatives) There exist c1,c2>0c_1,c_2>0 such that for all (y,η)(y,η), |s˙η(y,η)|≤c1σ2and|s¨η(y,η)|≤c2σ2.| s_η(y,η)|≤ c_1σ^2 | s_η(y,η)|≤ c_2σ^2. (i) (Sub-exponential tail) ‖sη(Y,η)‖ψ1≤Cψ1\|s_η(Y,η)\|_ _1≤ C_ _1, where Cψ1≍1/σC_ _1 1/σ. Remark C.4 (BTL verification). For the BTL model, s˙=−p(1−p)/σ2 s=-p(1-p)/σ^2 gives c1=1/4c_1=1/4; s¨=p(1−p)(1−2p)/σ3≤C/σ2 s=p(1-p)(1-2p)/σ^3≤ C/σ^2 (since σ is bounded below); s=(Y−p)/σs=(Y-p)/σ is bounded in [−1/σ,1/σ][-1/σ,1/σ], giving Cψ1≍1/σC_ _1 1/σ. C.2.5 Operators and directions Define the (non-constant weight) information operator G:=⋆[I(η⋆(X))(X⊗X)],so that⟨H1,GH2⟩=⋆[I(η⋆(X))⟨H1,X⟩⟨H2,X⟩].G:=E \! [\,I(η (X))\ (X X)\, ], that H_1,GH_2 =E \! [I(η (X))\, H_1,X \, H_2,X ]. (C.9) Equivalently, ⟨GU,V⟩=−ℙ⋆[s˙η(Y,η⋆(X))⟨U,X⟩⟨V,X⟩] GU,V =-P [ s_η(Y,η (X)) U,X V,X ], since [s˙η(Y,η⋆)|X]=−I(η⋆)E[ s_η(Y,η )|X]=-I(η ). Let P_T denote the orthogonal projector onto the tangent space T at T⋆T , and P P_T the estimated tangent projector built from estimated subspaces U^j\ U_j\. Define the restricted information operators A:=PGP,A^:=P^GP^,A:=P_T\,G\,P_T, A:= P_T\,G\, P_T, both invertible on their respective ranges. The oracle and estimated EIF directions are H⋆:=A−1PΓ,H^:=A^−1P^Γ.H :=A^-1P_T , H:= A^-1 P_T\, . (C.10) C.2.6 Γ coefficient vector Expand the functional gradient Γ∈ℋ in the canonical basis: Γ=∑ωγωEω,‖Γ‖1=∑ω|γω|. = _ω _ωE_ω, _1= _ω| _ω|. (C.11) C.2.7 Column-sum-zero constraint and block projections In the comparison model (mode 1), the orthogonal complement on mode 1 is Q1−PU1Q_1-P_U_1 where Q1:=Id1−1d1d1d1⊤Q_1:=I_d_1- 1d_11_d_11_d_1 (projection onto column-sum-zero subspace). However, since all tensors in our analysis already satisfy column-sum-zero (i.e., Q1H=HQ_1H=H for all relevant H), we have Q1=IQ_1=I on the relevant subspace, so Q1−PU1=I−PU1Q_1-P_U_1=I-P_U_1 and all block-projection formulas remain unchanged. See Section C.14.3 for details on the block decomposition. C.2.8 Operator norm assumptions We collect the key assumption on operator norms used throughout. Assumption C.5 (Inverse scaling). There exists an absolute constant CA>0C_A>0 such that ‖A^−1‖∞→∞∨‖A−1‖∞→∞≤CAσ2d⋆. A^-1 _∞→∞\ \ A^-1 _∞→∞\ ≤\ C_A\,σ^2d . Under the constant-weight BTL baseline, A0=P/(2σ2d⋆)A_0=P_T/(2σ^2d ) so ‖A0−1‖∞→∞=2σ2d⋆‖P‖∞→∞ A_0^-1 _∞→∞=2σ^2d \, P_T _∞→∞, giving CA=2‖P‖∞→∞=poly(μ,r,m)C_A=2\, P_T _∞→∞=poly(μ,r,m). Remark C.6. Since ‖P‖∞→∞≤poly(μ,r,m) P_T _∞→∞ (μ,r,m) (Lemma C.36) and ‖G‖∞→∞≤c+/(σ2d⋆) G _∞→∞≤ c_+/(σ^2d ) (from ‖G‖∞→∞≍1/(σ2d⋆) G _∞→∞ 1/(σ^2d ) as it’s a near-identity mapping on the full space, we have ‖A−1PG‖∞→∞ A^-1P_T\,G _∞→∞ ≤‖A−1‖∞→∞⋅‖P‖∞→∞⋅‖G‖∞→∞ \;≤\; A^-1 _∞→∞· P_T _∞→∞· G _∞→∞ ≤CAσ2d⋆⋅poly(μ,r,m)⋅c+σ2d⋆=CA⋅poly(μ,r,m). \;≤\;C_A\,σ^2d ·poly(μ,r,m)· c_+σ^2d \;=\;C_A·poly(μ,r,m). C.3 Proof strategy and norm scales This appendix is organized as follows. Section C.1 establishes that the initial estimator in Assumption 4.1, together with bounded signal and sample size conditions, automatically yields the subspace perturbation properties (P3)–(P5) needed for the proof. The remainder of this appendix proves the one-step estimator remainder bound (Theorem 4.4). We begin with a high-level overview of the proof and a collection of norm estimates that will be used repeatedly. C.3.1 Proof strategy: true G first, then G G perturbation In practice, the estimator uses the estimated information operator G G (built from Fisher weights I(η^)I( η) evaluated at the plug-in estimate). However, for clarity of exposition, we carry out the entire five-term remainder analysis in Sections C.4–C.9 as if the true operator G were used, i.e., we set A^:=P^GP A:= P_TG P_T with the true G. Section C.10 then shows that replacing G by the plug-in estimate G G (with Fisher weights I(η^)I( η) in place of I(η⋆)I(η )) introduces only a multiplicative (1+O(‖Δ‖∞))(1+O( _∞)) correction, which is negligible. This two-stage structure—oracle analysis followed by perturbation accounting—simplifies the exposition considerably and isolates the source of each error term. C.3.2 Norm scales for tangent-space projections The following norm estimates for the tangent-space projector applied to a basis tensor EωE_ω appear throughout the proof. They quantify the “spread” of a single coordinate direction after projection onto the tangent space, and are the key to understanding dimension cancellations in the error bounds. Lemma C.7 (Norm scales for PEωP_TE_ω). Under μ-incoherence, for every canonical basis tensor EωE_ω: ‖PEω‖∞ P_TE_ω _∞ ≤C(μ,r,m)d¯d⋆, ≤ C(μ,r,m)\, dd , (C.12) ‖PEω‖F P_TE_ω _F ≤C(μ,r,m)d¯d⋆, ≤ C(μ,r,m)\, dd , ‖PEω‖1 P_TE_ω _1 ≤C(μ,r,m)(dimension-free). ≤ C(μ,r,m) (dimension-free). The same bounds hold for P^Eω P_TE_ω under the estimated incoherence guarantee (P4). Intuition. The tangent-space projector P_T is a sum of m+1m+1 Kronecker block projectors ΠS _S with |S|≤1|S|≤ 1. The “core” block Π∅ _ compresses all modes, giving an ℓ∞ _∞ contribution of order ∏j(μrj/dj)≍μm∏jrj/d⋆ _j(μ r_j/d_j) μ^m _jr_j/d ; each “arm” block Πk _\k\ leaves mode k uncompressed (contributing a factor of ∼1 1 rather than μrk/dk μ r_k/d_k in that mode), which costs an extra dk/(μrk)≍d¯ d_k/(μ r_k) d relative to the core. Summing over blocks: • ℓ∞ _∞: dominated by the arm blocks, giving d¯/d⋆ d/d . • Frobenius: since ‖PEω‖F2=(P)ωω≤‖PEω‖∞≤Cd¯/d⋆ P_TE_ω _F^2=(P_T)_ω≤ P_TE_ω _∞≤ C d/d , the Frobenius norm is d¯/d⋆ d/d . • ℓ1 _1: each single-mode factor has ℓ1 _1 norm μrj μ r_j (bounded by Cauchy–Schwarz and incoherence), so the Kronecker product gives ∏jμrj=poly(μ,r,m) _j μ r_j=poly(μ,r,m)—dimension-free. Remark C.8 (Effect of subspace estimation error). When P_T is replaced by P^−P P_T-P_T, each norm picks up an additional factor of ρ: ‖(P^−P)Eω‖∞ ( P_T-P_T)E_ω _∞ ≤C(μ,r,m)d¯d⋆⋅ρ, ≤ C(μ,r,m)\, dd ·ρ, ‖(P^−P)Eω‖F ( P_T-P_T)E_ω _F ≤C(μ,r,m)d¯d⋆⋅ρ, ≤ C(μ,r,m)\, d d ·ρ, ‖(P^−P)Eω‖1 ( P_T-P_T)E_ω _1 ≤C(μ,r,m)⋅ρ. ≤ C(μ,r,m)·ρ. Here ρ:=σλmind⋆d¯log⁡d¯nρ:= σ _ d d dn is the subspace estimation parameter. The factor ρ arises from the spectral perturbation bound on ‖PU^j−PUj‖2 P_ U_j-P_U_j _2 (Eq. (C.17)). The Frobenius bound for the tangent projector difference also acquires an extra d¯ d beyond ρ/d⋆ρ/ d from the uncompressed mode in the arm blocks of P_T; see (C.18). The ℓ1 _1 dimension cancellation. A recurring theme in the bias analysis is the interplay between ‖A−1‖1→1≍σ2d⋆\|A^-1\|_1→ 1 σ^2d and the comparison-design averaging ⋆[|⟨v,X⟩|]≤‖v‖1/d⋆E [| v,X |]≤ v _1/d . Because ‖PEω‖1=O(1) P_TE_ω _1=O(1) (dimension-free), the product ‖A−1‖1→1⋅‖PEω‖1d⋆=CAσ2d⋆⋅O(1)d⋆=CAσ2⋅O(1) \|A^-1\|_1→ 1· P_TE_ω _1d \;=\; C_A\,σ^2d · O(1)d \;=\;C_A\,σ^2· O(1) is dimension-free (up to CAC_A and the noise level). This cancellation is the key mechanism behind the dimension-free bias bounds in Propositions C.16–C.20. We now proceed to bound each remainder term individually. C.4 Bounding RempHR_emp^H (Direction-Error Empirical Process) Intuition. This term captures the stochastic fluctuation from using the estimated direction H H in place of the oracle direction H⋆H . The key challenge is that H^−H⋆ H-H depends on the inverse restricted information operator A−1A^-1, which introduces the CAC_A factor. The proof follows a five-step template: (1) reduce the variance to a Frobenius norm via the pairwise-comparison second-moment identity; (2–3) extract the ‖Γ‖1 _1 factor and bound the per-basis-element Frobenius error using subspace perturbation; (4) bound the sub-exponential norm via ℓ∞ _∞ control, where CAC_A enters; (5) combine via Bernstein’s inequality. Recall RempH:=(ℙn−ℙ⋆)[sη(Y,η^)⟨H^−H⋆,X⟩].R_emp^H:=(P_n-P ) [s_η(Y, η)\, H-H ,X ]. Let Zi:=sη(Yi,η^i)⟨H^−H⋆,Xi⟩Z_i:=s_η(Y_i, η_i)\, H-H ,X_i , so RempH=1n∑i=1n(Zi−⋆[Zi])R_emp^H= 1n _i=1^n(Z_i-E [Z_i]). We bound its variance (Step 1), extract ‖Γ‖1 _1 (Steps 2–3), bound the ψ1 _1 norm (Step 4), and combine via Bernstein (Step 5). C.4.1 Step 1: Variance bound via Frobenius reduction By the Fisher comparability in Assumption C.3(i), ⋆[Zi2]=⋆[sη(Y,η^)2⟨H^−H⋆,X⟩2]≤C02⋆[I(η⋆)⟨H^−H⋆,X⟩2],E [Z_i^2]=E \! [s_η(Y, η)^2\, H-H ,X ^2 ]≤ C_0^2\,E \! [I(η )\, H-H ,X ^2 ], where C0C_0 is bounded by a function of c−,c+c_-,c_+ when T T is close to T⋆T . Applying Corollary C.32 with H=H^−H⋆H= H-H : Var(Zi)≤⋆[Zi2]≤C02c+σ2⋅‖H^−H⋆‖F2d⋆.Var(Z_i) [Z_i^2]≤ C_0^2\,c_+σ^2· H-H _F^2d . (C.13) This is the key: the Frobenius reduction (Lemma C.31) gives a 1/d⋆1/d factor. C.4.2 Step 2: Extracting ‖Γ‖1 _1 from the Frobenius norm Since H^−H⋆=ℳΓ H-H =M with ℳ:=A^−1P^−A−1PM:= A^-1 P_T-A^-1P_T, by Lemma C.33: ‖ℳΓ‖F≤‖Γ‖1⋅maxω⁡‖ℳEω‖F. _F≤ _1· _ω _ω _F. (C.14) Therefore Var(Zi)≤Cσ2⋅‖Γ‖12d⋆⋅(maxω⁡‖ℳEω‖F)2.Var(Z_i)≤ Cσ^2· _1^2d · ( _ω _ω _F )^2. (C.15) C.4.3 Step 3: Bounding maxω⁡‖ℳEω‖F _ω _ω _F Decompose ℳM via the resolvent identity: ℳ=A−1(P^−P)+A−1(A−A^)A^−1P^.M=A^-1( P_T-P_T)+A^-1(A- A) A^-1 P_T. (C.16) Term 1: ‖A−1(P^−P)Eω‖F A^-1( P_T-P_T)E_ω _F. By submultiplicativity, ‖A−1(P^−P)Eω‖F≤‖A−1‖op⋅‖(P^−P)Eω‖F. A^-1( P_T-P_T)E_ω _F≤ A^-1 _op· ( P_T-P_T)E_ω _F. From Assumption C.5, ‖A−1‖op≤‖A−1‖∞→∞≤CAσ2d⋆ A^-1 _op≤ A^-1 _∞→∞≤ C_Aσ^2d . For ‖(P^−P)Eω‖F ( P_T-P_T)E_ω _F, we use the telescope identity across modes: P^−P P_T-P_T decomposes as a sum of m terms, each involving (PU^j−PUj)(P_ U_j-P_U_j) in one mode and incoherent projections in the remaining modes. By the subspace estimation guarantee: ‖PU^j−PUj‖2≤C1ρμrj/dj,ρ:=σλmind⋆d¯log⁡d¯n. P_ U_j-P_U_j _2≤ C_1ρ μ r_j/d_j, ρ:= σ _ d d dn. (C.17) Combined with incoherence (‖Uk‖2,∞≤μrk/dk U_k _2,∞≤ μ r_k/d_k) in each remaining mode. However, the tangent projector P_T contains blocks where (I−PUk)(I-P_U_k) acts in one mode. When (I−PUk)(I-P_U_k) is applied to a standard basis vector eike_i_k, its norm is bounded by 11 (not μrk/dk μ r_k/d_k), contributing an extra factor dk/(μrk)≍d¯ d_k/(μ r_k) d. Accounting for this and summing over the m telescope terms: ‖(P^−P)Eω‖F≤C(μ,r,m)⋅d¯ρd⋆. ( P_T-P_T)E_ω _F≤ C(μ,r,m)· d\,ρ d . (C.18) Multiplying by ‖A−1‖op≤CAσ2d⋆ A^-1 _op≤ C_Aσ^2d : ‖A−1(P^−P)Eω‖F≤C(μ,r,m)CAσ2d⋆d¯⋅ρ. A^-1( P_T-P_T)E_ω _F≤ C(μ,r,m)\,C_A\,σ^2 d d·ρ. (C.19) Term 2: ‖A−1(A−A^)A^−1P^Eω‖F A^-1(A- A) A^-1 P_TE_ω _F. Using A−A^=(P−P^)GP+P^G(P−P^)A- A=(P_T- P_T)GP_T+ P_TG(P_T- P_T), ‖G‖op≤c+/(σ2d⋆) G _op≤ c_+/(σ^2d ), and ‖P^Eω‖F≤C(μ,r,m)/d⋆ P_TE_ω _F≤ C(μ,r,m)/ d (incoherence): ‖A−1(A−A^)A^−1P^Eω‖F≤C(μ,r,m)CA2σ2d⋆d¯⋅ρ. A^-1(A- A) A^-1 P_TE_ω _F≤ C(μ,r,m)\,C_A^2\,σ^2 d d·ρ. (C.20) This is the same order as Term 1. Combining. maxω⁡‖ℳEω‖F≤C(μ,r,m)CAσ2d⋆d¯⋅ρ. _ω _ω _F≤ C(μ,r,m)\,C_A\,σ^2 d d·ρ. (C.21) Substituting into (C.15): Var(Zi)≤C(μ,r,m)CA2σ2d¯‖Γ‖12ρ2.Var(Z_i)≤ C(μ,r,m)\,C_A^2\,σ^2\, d\, _1^2\,ρ^2. (C.22) C.4.4 Step 4: Sub-exponential (ψ1 _1) bound via ℓ∞ _∞ norm For the tail term, we need to bound ‖Zi‖ψ1 Z_i _ _1. By Assumption C.3(i), ‖sη(Y,η^)‖ψ1≤Cψ1 s_η(Y, η) _ _1≤ C_ _1, where Cψ1≍1/σC_ _1 1/σ. Since |⟨H^−H⋆,X⟩|≤2‖H^−H⋆‖∞| H-H ,X |≤ 2 H-H _∞ (comparison atom): ‖Zi‖ψ1≤2Cψ1⋅‖H^−H⋆‖∞. Z_i _ _1≤ 2C_ _1· H-H _∞. (C.23) By the ℓ∞ _∞ version of Lemma C.33: ‖H^−H⋆‖∞=‖ℳΓ‖∞≤‖Γ‖1⋅maxω⁡‖ℳEω‖∞. H-H _∞= _∞≤ _1· _ω _ω _∞. To bound ‖ℳEω‖∞ _ω _∞, we use the same decomposition (C.16) and submultiplicativity in ∥⋅∥∞→∞ · _∞→∞: ‖ℳEω‖∞≤‖A−1‖∞→∞⋅‖PEω‖∞+‖A−1‖∞→∞⋅‖A−A^‖∞→∞⋅‖A^−1‖∞→∞⋅‖P^Eω‖∞. _ω _∞≤ A^-1 _∞→∞· P_TE_ω _∞+ A^-1 _∞→∞· A- A _∞→∞· A^-1 _∞→∞· P_TE_ω _∞. By incoherence (Lemma C.36): ‖PEω‖∞,‖P^Eω‖∞≤C(μ,r,m)d¯/d⋆. P_TE_ω _∞,\; P_TE_ω _∞≤ C(μ,r,m)\, d/d . The second term is lower order under ‖P^−P‖≪1 P_T-P_T 1, so the leading contribution is maxω⁡‖ℳEω‖∞≤C(μ,r,m)⋅‖A−1‖∞→∞⋅d¯d⋆. _ω _ω _∞≤ C(μ,r,m)· A^-1 _∞→∞· dd . (C.24) Now substitute the key quantities: • ‖A−1‖∞→∞≤CAσ2d⋆ A^-1 _∞→∞≤ C_A\,σ^2d (Assumption C.5), • Cψ1≍1/σC_ _1 1/σ (Assumption C.3(i)). The d⋆d in ‖A−1‖∞→∞ A^-1 _∞→∞ cancels the 1/d⋆1/d from incoherence: ‖A−1‖∞→∞⋅d¯d⋆=CAσ2d⋆⋅d¯d⋆=CAσ2d¯. A^-1 _∞→∞· dd =C_A\,σ^2d · dd =C_A\,σ^2\, d. Multiplying by Cψ1≍1/σC_ _1 1/σ: ‖Zi‖ψ1≤C(μ,r,m)CAσd¯‖Γ‖1. Z_i _ _1≤ C(μ,r,m)\,C_A\,σ\, d\, _1. (C.25) C.4.5 Step 5: Bernstein concentration By Bernstein’s inequality (Lemma C.39 for sub-exponential variables), with probability at least 1−δ1-δ (conditional on the first-stage data): |RempH|≤C(Var(Zi)log⁡(2/δ)n+‖Zi‖ψ1log⁡(2/δ)n).|R_emp^H|≤ C ( Var(Z_i)\, (2/δ)n+ Z_i _ _1\, (2/δ)n ). (C.26) Substituting (C.22) and (C.25): Theorem C.9 (Bound on RempHR_emp^H). Under Assumptions C.3 and C.5, and the subspace estimation guarantees, with probability at least 1−δ1-δ (conditional on 1D_1): |RempH|≤C(μ,r,m)CA∥Γ∥1[σd¯ρlog⁡(2/δ)n+σd¯log⁡(2/δ)n], |R_emp^H|≤ C(μ,r,m)\,C_A\, _1 [σ\, d\;ρ\; (2/δ)n\;+\;σ\, d\; (2/δ)n ], (C.27) where ρ=σλmind⋆d¯log⁡d¯nρ= σ _ d d dn. Proof of Theorem C.9. The variance term gives Var(Zi)log⁡(2/δ)n≤C(μ,r,m)CAσd¯‖Γ‖1ρlog⁡(2/δ)n. Var(Z_i) (2/δ)n≤ C(μ,r,m)\,C_A\,σ\, d\, _1\,ρ\, (2/δ)n. The sub-exponential term gives (from (C.25)): ‖Zi‖ψ1log⁡(2/δ)n≤C(μ,r,m)CAσd¯‖Γ‖1log⁡(2/δ)n. Z_i _ _1 (2/δ)n≤ C(μ,r,m)\,C_A\,σ\, d\, _1\, (2/δ)n. Combining yields (C.27). □ Remark C.10 (Simplified pairwise-comparison bound). In the pairwise-comparison setting (σ=O(1)σ=O(1), λmin≍d⋆ _ d ), substituting ρ≍d¯logc⁡d¯/nρ d ^c d/n and δ=d¯−cδ= d^-c, the bound (C.27) simplifies to |RempH|≤C(μ,r,m)CA‖Γ‖1d¯logc⁡d¯n|R_emp^H|\;≤\;C(μ,r,m)\,C_A\, _1\, d ^c dn with probability ≥1−d¯−c≥ 1- d^-c. The σ2/λminσ^2/ _ ratio and d⋆d factors have been absorbed, leaving a clean d¯/n d/n rate multiplied by CAC_A. C.5 Bounding RempηR_emp^η (Score-Perturbation Empirical Process) Intuition. This term measures the stochastic error from evaluating the score at η η instead of η⋆η . A Taylor expansion of the score yields a first-order term (linear in Δ ) and a second-order term (quadratic in Δ ). Both are controlled via Bernstein’s inequality. The bounds on ‖H⋆‖∞\|H \|_∞ and ‖H⋆‖F/d⋆\|H \|_F/ d carry the CAC_A factor through the direction H⋆=A−1PΓH =A^-1P_T . In the pairwise-comparison setting, the σ2σ^2 factors from A−1A^-1 cancel with the 1/σ21/σ^2 from the score derivatives. Recall Rempη:=(ℙn−ℙ⋆)[(sη(Y,η^)−sη(Y,η⋆))⟨H⋆,X⟩].R_emp^η:=(P_n-P ) [(s_η(Y, η)-s_η(Y,η ))\, H ,X ]. Condition on the first-stage output so that Δ:=T^−T⋆ := T-T and H⋆H are fixed. C.5.1 Taylor expansion of the score For each i there exists η~i η_i between ⟨T^,Xi⟩ T,X_i and ⟨T⋆,Xi⟩ T ,X_i such that sη(Yi,η^i)−sη(Yi,ηi⋆)=s˙η(Yi,ηi⋆)⟨Δ,Xi⟩+12s¨η(Yi,η~i)⟨Δ,Xi⟩2.s_η(Y_i, η_i)-s_η(Y_i,η _i)= s_η(Y_i,η _i) ,X_i + 12 s_η(Y_i, η_i) ,X_i ^2. Hence Rempη=Rη,1emp+Rη,2empR_emp^η=R_η,1^emp+R_η,2^emp where Rη,1emp:=(ℙn−ℙ⋆)[s˙η(Y,η⋆)⟨Δ,X⟩⟨H⋆,X⟩],Rη,2emp:=12(ℙn−ℙ⋆)[s¨η(Y,η~)⟨Δ,X⟩2⟨H⋆,X⟩].R_η,1^emp:=(P_n-P ) [ s_η(Y,η ) ,X H ,X ], R_η,2^emp:= 12(P_n-P ) [ s_η(Y, η) ,X ^2 H ,X ]. C.5.2 Comparison-design moment bounds Under the comparison design: • |⟨Δ,X⟩|=|Δu,a−Δu,b|≤2‖Δ‖∞| ,X |=| _u,a- _u,b|≤ 2 _∞. • |⟨H⋆,X⟩|≤2‖H⋆‖∞| H ,X |≤ 2 H _∞. • ⋆[⟨H⋆,X⟩2]=‖H⋆‖F2/d⋆E [ H ,X ^2]= H _F^2/d (by Lemma C.31). C.5.3 Abstract Bernstein bound Proposition C.11 (Abstract bound for RempηR_emp^η). Under Assumption C.3 (with |s˙η|≤c1/σ2| s_η|≤ c_1/σ^2, |s¨η|≤c2/σ2| s_η|≤ c_2/σ^2) and comparison design, with probability at least 1−δ1-δ, |Rempη|≤ |R_emp^η|\;≤\; Cσ2[∥Δ∥∞‖H⋆‖Fd⋆log⁡(4/δ)n+∥Δ∥∞∥H⋆∥∞log⁡(4/δ)n Cσ^2 [ _∞\, H _F d (4/δ)n\;+\; _∞\, H _∞\, (4/δ)n +∥Δ∥∞2‖H⋆‖Fd⋆log⁡(4/δ)n+∥Δ∥∞2∥H⋆∥∞log⁡(4/δ)n], +\; _∞^2\, H _F d (4/δ)n\;+\; _∞^2\, H _∞\, (4/δ)n ], (C.28) where C absorbs c1,c2c_1,c_2. Proof of Proposition C.11. We bound each order using Bernstein’s inequality (Lemma C.39). For the first-order term Rη,1empR_η,1^emp, define Wi(1):=s˙η(Yi,ηi⋆)⟨Δ,Xi⟩⟨H⋆,Xi⟩−⋆[⋅].W_i^(1):= s_η(Y_i,η _i) ,X_i H ,X_i -E [·]. Then |Wi(1)|≤8(c1/σ2)∥Δ∥∞∥H⋆∥∞=:M1|W_i^(1)|≤ 8(c_1/σ^2) _∞ H _∞=:M_1, and Var(Wi(1))≤4c12σ4‖Δ‖∞2‖H⋆‖F2d⋆.Var(W_i^(1))≤ 4c_1^2σ^4\, _∞^2\, H _F^2d . For the second-order term Rη,2empR_η,2^emp, similarly |Wi(2)|≤4(c2/σ2)‖Δ‖∞2‖H⋆‖∞,Var(Wi(2))≤4c22σ−4‖Δ‖∞4‖H⋆‖F2/d⋆.|W_i^(2)|≤ 4(c_2/σ^2) _∞^2 H _∞, (W_i^(2))≤ 4c_2^2σ^-4 _∞^4 H _F^2/d . Applying Bernstein to each term and taking a union bound yields (C.28). □ C.5.4 Bounds on ‖H⋆‖∞ H _∞ and ‖H⋆‖F/d⋆ H _F/ d By Lemma C.33 and Assumption C.5: ℓ∞ _∞ bound. Using ‖PEω‖∞≤Cμd¯/d⋆ P_TE_ω _∞≤ C_μ\, d/d (Lemma C.37, with d¯:=maxj⁡dj d:= _jd_j): ‖H⋆‖∞≤‖Γ‖1⋅‖A−1‖∞→∞⋅maxω⁡‖PEω‖∞≤C(μ,r,m)CAσ2d¯‖Γ‖1. H _∞≤ _1· A^-1 _∞→∞· _ω P_TE_ω _∞≤ C(μ,r,m)\,C_A\,σ^2 d\, _1. (C.29) Frobenius bound. Since ‖PEω‖F2=(P)ωω≤‖PEω‖∞≤Cμd¯/d⋆,‖A−1‖op≤‖A−1‖∞→∞≤CAσ2d⋆, P_TE_ω _F^2=(P_T)_ω≤ P_TE_ω _∞≤ C_μ\, d/d , A^-1 _op≤ A^-1 _∞→∞≤ C_A\,σ^2d , we obtain: ‖H⋆‖Fd⋆≤‖Γ‖1⋅‖A−1‖opmaxω⁡‖PEω‖Fd⋆≤C(μ,r,m)CAσ2d¯‖Γ‖1. H _F d ≤ _1· A^-1 _op _ω P_TE_ω _F d ≤ C(μ,r,m)\,C_A\,σ^2 d\, _1. (C.30) C.5.5 Final explicit bound Substituting (C.29) and (C.30) into Proposition C.11. In each term, the factor 1/σ21/σ^2 from Assumption C.3 cancels with σ2σ^2 from ‖A−1‖ A^-1 : Corollary C.12 (Explicit bound for RempηR_emp^η). Under Assumptions C.3 and C.5, with probability at least 1−δ1-δ, |Rempη|≤C(μ,r,m)CA‖Γ‖1(‖Δ‖∞+‖Δ‖∞2)[d¯log⁡(4/δ)n+d¯log⁡(4/δ)n].|R_emp^η|\;≤\;C(μ,r,m)\,C_A\, _1 ( _∞+ _∞^2 ) [ d (4/δ)n\;+\; d\, (4/δ)n ]. (C.31) Proof of Corollary C.12. The four terms in (C.28) become, after substitution: the first-order variance term gives 1σ2‖Δ‖∞‖H⋆‖Fd⋆≤CCA‖Γ‖1‖Δ‖∞d¯ 1σ^2 _∞ H _F d ≤ C\,C_A\, _1\, _∞ d (since σ2σ^2 cancels); the first-order tail term gives 1σ2‖Δ‖∞‖H⋆‖∞≤CCA‖Γ‖1‖Δ‖∞d¯ 1σ^2 _∞ H _∞≤ C\,C_A\, _1\, _∞\, d (again σ2σ^2 cancels); and the second-order variance and tail terms are the same as their first-order counterparts with an extra factor of ‖Δ‖∞ _∞. Factoring ‖Δ‖∞+‖Δ‖∞2 _∞+ _∞^2 and collecting the 1/n 1/n and 1/n1/n terms yields (C.31). □ C.6 Bounding RprojR_proj (Projection Leakage) Intuition. The projection leakage captures the bias from the component of Γ orthogonal to the tangent space. Since Δ=T^−T⋆ = T-T has its normal component (I−P)Δ(I-P_T) supported on blocks ΠS _S with |S|≥2|S|≥ 2, this term is quadratic in subspace estimation errors. This is the key structural reason why projection leakage is lower order: it requires errors in two or more modes simultaneously. The projection leakage term is Rproj:=⟨(I−P)Γ,Δ⟩.R_proj:= (I-P_T) , . C.6.1 Basic bound By Hölder’s inequality, |Rproj|≤‖Γ‖1⋅‖(I−P)Δ‖∞.|R_proj|≤ _1· (I-P_T) _∞. (C.32) The key structural fact (Lemma C.35) is that (I−P)Δ=∑S⊆[m]|S|≥2ΠS(Δ),(I-P_T) = _ subarraycS [m]\\ |S|≥ 2 subarray _S( ), so the projection leakage is supported on blocks ΠS _S with |S|≥2|S|≥ 2, making it quadratic in subspace estimation errors. C.6.2 The ε2 ^2 bound on ‖(I−P^)T^‖∞ (I- P_T) T _∞ Define εj:=‖PU^j,0−PUj‖2,∞ _j:= P_ U_j,0-P_U_j _2,∞ (projector error in the j-th mode). Proposition C.13 (Complete ε2 ^2 argument). Let T^=C^×1U^1,0⋯×mU^m,0 T= C×_1 U_1,0·s×_m U_m,0. Then ‖(I−P^)T^‖∞≤‖C^‖op(∏k=1m(‖U^k,0‖2,∞+εk))∑S⊆[m]|S|≥2∏j∈Sεj. (I- P_T) T _∞≤ C _op ( _k=1^m( U_k,0 _2,∞+ _k) ) _ subarraycS [m]\\ |S|≥ 2 subarray _j∈ S _j. (C.33) On the event maxj⁡εj≤1 _j _j≤ 1, ∑S⊆[m]|S|≥2∏j∈Sεj≤2m−2∑1≤a<b≤mεaεb, _ subarraycS [m]\\ |S|≥ 2 subarray _j∈ S _j≤ 2^m-2 _1≤ a<b≤ m _a _b, (C.34) so ‖(I−P^)T^‖∞ (I- P_T) T _∞ is quadratic in the subspace errors. Proof of Proposition C.13. By I−P^=∑|S|≥2Π^SI- P_T= _|S|≥ 2 _S and the triangle inequality, ‖(I−P^)T^‖∞≤∑|S|≥2‖Π^S(T^)‖∞. (I- P_T) T _∞≤ _|S|≥ 2 _S( T) _∞. Fix S with |S|≥2|S|≥ 2. Using the Tucker form of T T, Π^S(T^)=C^×j∈SEj×k∉SMk, _S( T)= C×_j∈ SE_j×_k∉ SM_k, where Ej:=(I−PUj)U^j,0E_j:=(I-P_U_j) U_j,0 (error factor) and Mk:=PUkU^k,0M_k:=P_U_k U_k,0 (main factor). By the entrywise contraction lemma (Lemma C.40), ‖Π^S(T^)‖∞≤‖C^‖op∏j∈S‖Ej‖2,∞∏k∉S‖Mk‖2,∞. _S( T) _∞≤ C _op _j∈ S E_j _2,∞ _k∉ S M_k _2,∞. Since Ej=(PU^j,0−PUj)U^j,0E_j=(P_ U_j,0-P_U_j) U_j,0, we have ‖Ej‖2,∞≤εj E_j _2,∞≤ _j, and ‖Mk‖2,∞≤‖U^k,0‖2,∞+εk M_k _2,∞≤ U_k,0 _2,∞+ _k. Summing over all S with |S|≥2|S|≥ 2 yields (C.33). For the combinatorial bound (C.34), each pair a,b⊆[m]\a,b\ [m] is contained in exactly 2m−22^m-2 subsets S⊆[m]S [m], and since εj≤1 _j≤ 1, we have ∏j∈Sεj≤εaεb _j∈ S _j≤ _a _b for any S⊇a,bS \a,b\. □ C.6.3 Final bound under initialization guarantees We now specialize Proposition C.13 using the initialization guarantees from the main paper. Define the signal-to-noise ratio parameter ρ:=σλmind⋆d¯log⁡d¯n.ρ\;:=\; σ _ d \, d dn. (C.35) Under the SNR condition, ρ≪1ρ 1. By the subspace perturbation bounds (Lemma 5.1 of the main paper), the mode-j errors satisfy εj:=‖PU^j,0−PUj‖2,∞≤8C1σλminμrjdjd⋆d¯log⁡d¯n= 8C1ρμrjdj, _j\;:=\; P_ U_j,0-P_U_j _2,∞\;≤\; 8C_1σ _ μ r_jd_j d \, d dn\;=\;8C_1\,ρ\, μ r_jd_j, and the estimated subspaces are incoherent: ‖ea⊤U^j,0‖22≤2μrj/dj\|e_a U_j,0\|_2^2≤ 2μ r_j/d_j, so ‖U^j,0‖2,∞≤2μrj/dj U_j,0 _2,∞≤ 2μ r_j/d_j. Main factor. Since ρ≪1ρ 1, ‖U^k,0‖2,∞+εk≤(2+8C1ρ)μrkdk≤ 2μrkdk, U_k,0 _2,∞+ _k\;≤\; ( 2+8C_1ρ ) μ r_kd_k\;≤\;2 μ r_kd_k, hence ∏k=1m(‖U^k,0‖2,∞+εk)≤ 2mμmr⋆d⋆,r⋆:=∏k=1mrk. _k=1^m ( U_k,0 _2,∞+ _k )\;≤\;2^m\, μ^mr d , r := _k=1^mr_k. (C.36) Quadratic sum. Since μrj/dj≤1μ r_j/d_j≤ 1 (incoherence) and ρ≪1ρ 1, each εj≤8C1ρ≤1 _j≤ 8C_1ρ≤ 1. By (C.34), ∑S⊆[m]|S|≥2∏j∈Sεj≤ 2m−2∑a<bεaεb≤ 2m−2(8C1)2ρ2∑a<bμradaμrbdb≤C(μ,r,m)ρ2. _ subarraycS [m]\\ |S|≥ 2 subarray _j∈ S _j\;≤\;2^m-2 _a<b _a _b\;≤\;2^m-2(8C_1)^2\,ρ^2\! _a<b\! μ r_ad_a μ r_bd_b\;≤\;C(μ,r,m)\,ρ^2. (C.37) Core tensor factor. Since T^=C^×1U^1,0⋯×mU^m,0 T= C×_1 U_1,0·s×_m U_m,0 and T^≈T⋆ T≈ T , ‖C^‖opd⋆≤‖T^‖Fd⋆≤‖T^‖∞≤‖T⋆‖∞+o(1)=O(1), C _op d \;≤\; T _F d \;≤\; T _∞\;≤\; T _∞+o(1)\;=\;O(1), which is bounded by a constant depending only on the model parameters (bounded entries in the BTL model). Combining. Substituting (C.36) and (C.37) into Proposition C.13: ‖(I−P)Δ‖∞≤‖C^‖op⋅2mμmr⋆d⋆⋅C(μ,r,m)ρ2=C(μ,r,m)⋅‖C^‖opd⋆⋅ρ2≤C(μ,r,m)ρ2. (I-P_T) _∞\;≤\; C _op· 2^m μ^mr d · C(μ,r,m)\,ρ^2\;=\;C(μ,r,m)· C _op d ·ρ^2\;≤\;C(μ,r,m)\,ρ^2. By (C.32), |Rproj|≤ 2mC(μ,r,m)∥Γ∥1ρ2. |R_proj|\;≤\;2^m\,C(μ,r,m)\; _1\;ρ^2. (C.38) C.7 H-Direction Bias This term is the population-level error from using H H instead of H⋆H . After a Taylor expansion around η⋆η , the first-order piece involves ⟨GΔ,H^−H⋆⟩ G , H-H , which is controlled by the off-tangent components of Δ (since the tangent-space component cancels via the first-order cancellation mechanism in Section C.8). The second-order piece is bounded by CA‖Δ‖∞2‖Γ‖1C_A\| \|_∞^2\| \|_1 using the ℓ1 _1-average bound, where the key cancellation is d⋆d from ‖A−1‖1→1\|A^-1\|_1→ 1 against 1/d⋆1/d from the comparison averaging. The H-direction bias term is ℙ⋆[ST^(H^)−ST^(H⋆)]=ℙ⋆[sη(Y,η^)⟨H^−H⋆,X⟩].P [S_ T( H)-S_ T(H ) ]=P [s_η(Y, η) H-H ,X ]. C.7.1 Taylor expansion and leading term Expanding sη(Y,η^)s_η(Y, η) around η⋆η : ℙ⋆[sη(Y,η^)⟨H^−H⋆,X⟩] [s_η(Y, η) H-H ,X ] =ℙ⋆[sη(Y,η⋆)⟨H^−H⋆,X⟩]⏟= 0 (score mean-zero) = P [s_η(Y,η ) H-H ,X ]_=\,0 (score mean-zero) +ℙ⋆[s˙η(Y,η⋆)⟨Δ,X⟩⟨H^−H⋆,X⟩]⏟first-order in Δ + P [ s_η(Y,η ) ,X H-H ,X ]_first-order in +12ℙ⋆[s¨η(Y,η¯)⟨Δ,X⟩2⟨H^−H⋆,X⟩]⏟second-order in Δ. + 12P [ s_η(Y, η) ,X ^2 H-H ,X ]_second-order in . (C.39) C.7.2 ℓ1 _1-average bound under comparison design Lemma C.14 (ℓ1 _1-average bound). Under the comparison design, for any tensor v∈ℋv , ⋆[|⟨v,X⟩|]≤‖v‖1d⋆.E [| v,X | ]≤ v _1d . (C.40) Proof of Lemma C.14. Since X=eu,a−eu,bX=e_u,a-e_u,b with (u,a,b)(u,\a,b\) uniform, |⟨v,X⟩|=|vu,a−vu,b|≤|vu,a|+|vu,b|| v,X |=|v_u,a-v_u,b|≤|v_u,a|+|v_u,b|. Taking expectation, ⋆[|⟨v,X⟩|]≤1D(d12)∑u∑a<b(|vu,a|+|vu,b|)=1D(d12)(d1−1)∑u∑a|vu,a|=2Dd1‖v‖1.E [| v,X |]≤ 1D d_12 _u _a<b(|v_u,a|+|v_u,b|)= 1D d_12\,(d_1-1) _u _a|v_u,a|= 2Dd_1\, v _1. Since d⋆=D(d1−1)/2d =D(d_1-1)/2 and d1/(d1−1)≥1d_1/(d_1-1)≥ 1, 2Dd1=d1−1d1⋅1d⋆≤1d⋆ 2Dd_1= d_1-1d_1· 1d ≤ 1d . □ C.7.3 Dimension-free ‖PEω‖1 P_TE_ω _1 bound Lemma C.15 (Dimension-free ℓ1 _1 bound for tangent projection). Under μ-incoherence, for any canonical basis tensor Eω=ei1⊗⋯⊗eimE_ω=e_i_1 ·s e_i_m, ‖PEω‖1≤CP,CP:=C(μ,r,m). P_TE_ω _1≤ C_P, C_P:=C(μ,r,m). (C.41) The same bound holds for P P_T under the incoherence of the estimated subspaces ‖ea⊤U^j,0‖22≤2μrj/dj e_a U_j,0 _2^2≤ 2μ r_j/d_j. Proof of Lemma C.15. For elementary tensors, the Kronecker ℓ1 _1 factorization gives ‖a(1)⊗⋯⊗a(m)‖1=∏j‖a(j)‖1 a^(1) ·s a^(m) _1= _j a^(j) _1. By Cauchy–Schwarz and incoherence, each single-mode factor satisfies ‖Pjeij‖1=∑k|(Pj)k,ij|≤dj‖Pjeij‖2≤djμrj/dj=μrj, P_je_i_j _1= _k|(P_j)_k,i_j|≤ d_j\, P_je_i_j _2≤ d_j μ r_j/d_j= μ r_j, and ‖Pj⟂eij‖1≤‖eij‖1+‖Pjeij‖1≤1+μrj P_j e_i_j _1≤ e_i_j _1+ P_je_i_j _1≤ 1+ μ r_j. Using P=Π∅+∑kΠkP_T= _ + _k _\k\ and the Kronecker factorization, ‖Π∅(Eω)‖1=∏j‖Pjeij‖1≤μm/2r⋆ _ (E_ω) _1= _j P_je_i_j _1≤μ^m/2 r , and ‖Πk(Eω)‖1=‖Pk⟂eik‖1∏j≠k‖Pjeij‖1≤(1+μrk)∏j≠kμrj _\k\(E_ω) _1= P_k e_i_k _1 _j≠ k P_je_i_j _1≤(1+ μ r_k) _j≠ k μ r_j. Summing over blocks yields ‖PEω‖1≤C(μ,r,m) P_TE_ω _1≤ C(μ,r,m), which is dimension-free. The bound for P P_T follows identically with 2μ2μ replacing μ. □ C.7.4 Bounding the first-order piece Proposition C.16 (H-direction bias bound). Under Assumption C.5, the first-order piece satisfies |⟨GΔ,H^−H⋆⟩|≤C(μ,r,m)CA‖Γ‖1[‖(I−P^)Δ‖∞+‖(I−P)Δ‖∞]. | G , H-H |≤ C(μ,r,m)\,C_A\, _1 [ (I- P_T) _∞+ (I-P_T) _∞ ]. (C.42) Proof of Proposition C.16. By (C.9), ℙ⋆[s˙η(Y,η⋆)⟨Δ,X⟩⟨H^−H⋆,X⟩]=⟨GΔ,H^−H⋆⟩.P [ s_η(Y,η ) ,X H-H ,X ]= G , H-H . Since G is symmetric and A^−1P A^-1 P_T, A−1PA^-1P_T are self-adjoint, ⟨GΔ,H^−H⋆⟩=⟨(A^−1P^G−A−1PG)Δ,Γ⟩. G , H-H = ( A^-1 P_TG-A^-1P_TG) , . By Hölder’s inequality, |⟨u,Γ⟩|≤‖Γ‖1⋅‖u‖∞| u, |≤ _1· u _∞. Decomposing via the triangle inequality, ‖(A^−1P^G−A−1PG)Δ‖∞≤‖A^−1P^G(I−P^)Δ‖∞+‖A−1PG(I−P)Δ‖∞+‖(P^−P)Δ‖∞. ( A^-1 P_TG-A^-1P_TG) _∞≤ A^-1 P_TG(I- P_T) _∞+ A^-1P_TG(I-P_T) _∞+ ( P_T-P_T) _∞. Using ‖(P^−P)Δ‖∞≤‖(I−P^)Δ‖∞+‖(I−P)Δ‖∞ ( P_T-P_T) _∞≤ (I- P_T) _∞+ (I-P_T) _∞ and Remark C.6 yields (C.42). □ C.7.5 The second-order piece By |s¨η|≤c2/σ2| s_η|≤ c_2/σ^2 and |⟨Δ,X⟩|≤2‖Δ‖∞| ,X |≤ 2 _∞, |12ℙ⋆[s¨η⟨Δ,X⟩2⟨H^−H⋆,X⟩]|≤2c2σ2‖Δ‖∞2⋆[|⟨H^−H⋆,X⟩|]. | 12P [ s_η ,X ^2 H-H ,X ] |≤ 2c_2σ^2\, _∞^2\,E [| H-H ,X |]. We now bound ⋆[|⟨H^−H⋆,X⟩|]E [| H-H ,X |] using the ℓ1 _1-average (Lemma C.14): ⋆[|⟨H^−H⋆,X⟩|]≤‖H^−H⋆‖1d⋆.E [| H-H ,X | ]≤ H-H _1d . (C.43) Step 1: ℓ1 _1 extraction. By the ℓ1 _1-version of Lemma C.33, ‖H^−H⋆‖1=‖(A^−1P^−A−1P)Γ‖1≤‖Γ‖1maxω⁡‖(A^−1P^−A−1P)Eω‖1. H-H _1= ( A^-1 P_T-A^-1P_T) _1≤ _1\, _ω\, ( A^-1 P_T-A^-1P_T)E_ω _1. (C.44) Step 2: triangle inequality and symmetry. For each basis tensor EωE_ω, ‖(A^−1P^−A−1P)Eω‖1≤‖A^−1‖1→1‖P^Eω‖1+‖A−1‖1→1‖PEω‖1. ( A^-1 P_T-A^-1P_T)E_ω _1≤ A^-1 _1→ 1\, P_TE_ω _1+ A^-1 _1→ 1\, P_TE_ω _1. Since A^−1 A^-1 and A−1A^-1 are symmetric on their respective tangent spaces, Lemma C.38 gives ‖A^−1‖1→1=‖A^−1‖∞→∞≤CAσ2d⋆ A^-1 _1→ 1= A^-1 _∞→∞≤ C_A\,σ^2d (Assumption C.5), and likewise for A−1A^-1. By Lemma C.15, ‖PEω‖1≤CP P_TE_ω _1≤ C_P and ‖P^Eω‖1≤CP P_TE_ω _1≤ C_P. Therefore maxω⁡‖(A^−1P^−A−1P)Eω‖1≤2CAσ2d⋆⋅CP. _ω\, ( A^-1 P_T-A^-1P_T)E_ω _1≤ 2C_A\,σ^2d · C_P. (C.45) Step 3: assembling. Substituting (C.45) into (C.44): ‖H^−H⋆‖1≤2CACPσ2d⋆‖Γ‖1, H-H _1≤ 2C_AC_P\,σ^2d \, _1, so ‖H^−H⋆‖1d⋆≤2CACPσ2‖Γ‖1. H-H _1d ≤ 2C_AC_P\,σ^2\, _1. The second-order piece therefore satisfies |12ℙ⋆[s¨η⟨Δ,X⟩2⟨H^−H⋆,X⟩]|≤C(μ,r,m)CA∥Δ∥∞2∥Γ∥1. | 12P [ s_η ,X ^2 H-H ,X ] |≤ C(μ,r,m)\,C_A\, _∞^2\, _1. (C.46) Note the key cancellations: σ2σ^2 from ‖A−1‖1→1≍σ2d⋆ A^-1 _1→ 1 σ^2d cancels with 1/σ21/σ^2 from |s¨η|≤c2/σ2| s_η|≤ c_2/σ^2, and d⋆d from ‖A−1‖1→1 A^-1 _1→ 1 cancels with 1/d⋆1/d from the ℓ1 _1-average bound (Lemma C.14). Remark C.17. The first-order piece (Proposition C.16) is bounded by C(μ,r,m)CA‖Γ‖1[‖(I−P^)Δ‖∞+‖(I−P)Δ‖∞],C(μ,r,m)\,C_A\, _1 [ (I- P_T) _∞+ (I-P_T) _∞ ], where the off-tangent components are controlled by the ε2 ^2 bound of Proposition C.13. The second-order piece (C.46) is bounded by C(μ,r,m)CA‖Δ‖∞2‖Γ‖1,C(μ,r,m)\,C_A\, _∞^2\, _1, which is also O(ε2)O( ^2) since ‖Δ‖∞=O(ε) _∞=O( ). Both pieces retain explicit CAC_A dependence and are dimension-free up to ‖Γ‖1 _1. C.8 First-Order Cancellation This is the heart of the one-step estimator: the plug-in bias ⟨PΓ,Δ⟩ P_T , cancels exactly with the leading term of the population correction −⟨Δ,G(H⋆)⟩- ,G(H ) on the tangent space. The proof decomposes Δ=PΔ+(I−P)Δ =P_T +(I-P_T) : the tangent component cancels perfectly by the definition H⋆=A−1PΓH =A^-1P_T , while the normal component produces a residual controlled by ‖(I−P)Δ‖∞\|(I-P_T) \|_∞, which is quadratic in subspace errors. The combined first-order cancellation term from the bias decomposition is ⟨PΓ,Δ⟩+ℙ⋆[ST^(H⋆)−ST⋆(H⋆)]. P_T , +P [S_ T(H )-S_T (H ) ]. (C.47) C.8.1 Taylor expansion of the population score difference The second part of (C.47) is ℙ⋆[sη(Y,η^)⟨H⋆,X⟩]−ℙ⋆[sη(Y,η⋆)⟨H⋆,X⟩].P [s_η(Y, η) H ,X ]-P [s_η(Y,η ) H ,X ]. The second term vanishes ([sη|X]=0E[s_η|X]=0). Expanding the first around η⋆η : ℙ⋆[ST^(H⋆)−ST⋆(H⋆)] [S_ T(H )-S_T (H ) ] =ℙ⋆[s˙η(Y,η⋆)⟨Δ,X⟩⟨H⋆,X⟩]+12ℙ⋆[s¨η(Y,η¯)⟨Δ,X⟩2⟨H⋆,X⟩] =P [ s_η(Y,η ) ,X H ,X ]+ 12P [ s_η(Y, η) ,X ^2 H ,X ] =−⟨Δ,G(H⋆)⟩+R2(Δ;H⋆), =- ,G(H ) +R_2( ;H ), (C.48) where we used [s˙η(Y,η⋆)|X]=−I(η⋆)E[ s_η(Y,η )|X]=-I(η ) and defined R2(Δ;H⋆):=12ℙ⋆[s¨η(Y,η¯)⟨Δ,X⟩2⟨H⋆,X⟩].R_2( ;H ):= 12P [ s_η(Y, η) ,X ^2 H ,X ]. C.8.2 The cancellation mechanism Plugging (C.48) into (C.47): ⟨PΓ,Δ⟩−⟨Δ,G(H⋆)⟩+R2(Δ;H⋆). P_T , - ,G(H ) +R_2( ;H ). (C.49) Now decompose Δ=PΔ+(I−P)Δ =P_T +(I-P_T) in the first-order part: ⟨PΓ,Δ⟩−⟨Δ,G(H⋆)⟩ P_T , - ,G(H ) = = (⟨PΓ,PΔ⟩−⟨PΔ,G(H⋆)⟩) ( P_T ,P_T - P_T ,G(H ) ) + + (⟨PΓ,(I−P)Δ⟩−⟨(I−P)Δ,G(H⋆)⟩). ( P_T ,(I-P_T) - (I-P_T) ,G(H ) ). The tangent-space part cancels exactly. Since H⋆=A−1PΓH =A^-1P_T with A=PGPA=P_TGP_T: ⟨PΔ,G(H⋆)⟩=⟨PΔ,G(A−1PΓ)⟩=⟨PΔ,PG(A−1PΓ)⟩+⟨PΔ,(I−P)G(H⋆)⟩. P_T ,G(H ) = P_T ,G(A^-1P_T ) = P_T ,P_TG(A^-1P_T ) + P_T ,(I-P_T)G(H ) . The first term equals ⟨PΔ,A⋅A−1PΓ⟩=⟨PΔ,PΓ⟩, P_T ,A· A^-1P_T = P_T ,P_T , which cancels with ⟨PΓ,PΔ⟩ P_T ,P_T . The second term satisfies ⟨PΔ,(I−P)G(H⋆)⟩=0 P_T ,(I-P_T)G(H ) =0 by orthogonality. The normal component survives but is small. The surviving first-order term from (C.8.2) is ⟨PΓ,(I−P)Δ⟩−⟨(I−P)Δ,G(H⋆)⟩. P_T ,(I-P_T) - (I-P_T) ,G(H ) . (C.50) C.8.3 Bounding the surviving first-order term Proposition C.18. Under Assumption C.5, |⟨PΓ,Δ⟩−⟨Δ,G(H⋆)⟩|≤C‖Γ‖1‖(I−P)Δ‖∞, | P_T , - ,G(H ) |≤ C\, _1\, (I-P_T) _∞, (C.51) where C=CA⋅poly(μ,r,m)C=C_A·poly(μ,r,m) (Remark C.6). Proof of Proposition C.18. For the first part of (C.50), |⟨PΓ,(I−P)Δ⟩|=|⟨Γ,P(I−P)Δ⟩|=0| P_T ,(I-P_T) |=| ,P_T(I-P_T) |=0 since P(I−P)=0P_T(I-P_T)=0. For the second part, use symmetry of G and self-adjointness of A−1PA^-1P_T to transfer the operator onto Γ : ⟨(I−P)Δ,G(H⋆)⟩=⟨G(I−P)Δ,A−1PΓ⟩=⟨A−1PG(I−P)Δ,Γ⟩. (I-P_T) ,G(H ) = G(I-P_T) ,A^-1P_T = A^-1P_TG(I-P_T) , . By Hölder’s inequality, |⟨A−1PG(I−P)Δ,Γ⟩| | A^-1P_TG(I-P_T) , | ≤‖Γ‖1⋅‖A−1PG(I−P)Δ‖∞ ≤ _1· A^-1P_TG(I-P_T) _∞ ≤‖Γ‖1⋅‖A−1PG‖∞→∞⋅‖(I−P)Δ‖∞. ≤ _1· A^-1P_TG _∞→∞· (I-P_T) _∞. By Remark C.6, ‖A−1PG‖∞→∞=CA⋅poly(μ,r,m) A^-1P_TG _∞→∞=C_A·poly(μ,r,m), which gives (C.51). □ Remark C.19. Note ⟨PΓ,(I−P)Δ⟩=0 P_T ,(I-P_T) =0 exactly, so the surviving term is only −⟨(I−P)Δ,G(H⋆)⟩- (I-P_T) ,G(H ) , which is controlled by ‖(I−P)Δ‖∞ (I-P_T) _∞. By Proposition C.13, this is quadratic in subspace errors (O(ε2)O( ^2)), which is why only the second-order remainder R2R_2 survives as the dominant bias term. C.9 Second-Order Score Remainder Intuition. After the first-order cancellation, the dominant bias term is the second-order Taylor remainder R2(Δ;H⋆)=O(‖Δ‖∞2)R_2( ;H )=O(\| \|_∞^2). The key mechanism is the ℓ1 _1 dimension cancellation: ‖A−1PEω‖1/d⋆≤CAσ2⋅poly(μ,r,m)\|A^-1P_TE_ω\|_1/d ≤ C_Aσ^2·poly(μ,r,m), where the d⋆d in ‖A−1‖1→1≍σ2d⋆\|A^-1\|_1→ 1 σ^2d cancels with 1/d⋆1/d from the comparison averaging. This gives |R2|≤C(μ,r,m)CA‖Δ‖∞2‖Γ‖1|R_2|≤ C(μ,r,m)C_A\| \|_∞^2\| \|_1 without any residual dimension factor. The second-order score remainder from (C.48) is R2(Δ;H⋆)=12ℙ⋆[s¨η(Y,η¯)⟨Δ,X⟩2⟨H⋆,X⟩].R_2( ;H )= 12P [ s_η(Y, η) ,X ^2 H ,X ]. Proposition C.20 (Second-order remainder bound). Under Assumption C.3 (with |s¨η|≤c2/σ2| s_η|≤ c_2/σ^2) and comparison design, |R2(Δ;H⋆)|≤Cσ2‖Δ‖∞2‖H⋆‖1d⋆.|R_2( ;H )|≤ Cσ^2\, _∞^2\, H _1d . (C.52) Expanding H⋆=A−1PΓH =A^-1P_T and using ‖A−1‖∞→∞≤CAσ2d⋆ A^-1 _∞→∞≤ C_Aσ^2d , ‖P‖∞→∞=poly(μ,r,m) P_T _∞→∞=poly(μ,r,m): |R2(Δ;H⋆)|≤C(μ,r,m)CA‖Δ‖∞2‖Γ‖1.|R_2( ;H )|≤ C(μ,r,m)\,C_A\, _∞^2\, _1. (C.53) Proof of Proposition C.20. Since |⟨Δ,X⟩|≤2‖Δ‖∞| ,X |≤ 2 _∞ (comparison atom) and |s¨η|≤c2/σ2| s_η|≤ c_2/σ^2, |R2(Δ;H⋆)|≤2c2σ2‖Δ‖∞2⋆[|⟨H⋆,X⟩|].|R_2( ;H )|≤ 2c_2σ^2\, _∞^2\,E \! [| H ,X | ]. Under the comparison design, ⟨H⋆,X⟩=Hu,a⋆−Hu,b⋆ H ,X =H _u,a-H _u,b where (u,a,b)(u,\a,b\) is drawn uniformly, so by the triangle inequality and averaging, ⋆[|⟨H⋆,X⟩|]≤2d⋆‖H⋆‖1E \! [| H ,X | ]≤ 2d \, H _1, which gives (C.52) (absorbing the constant 44 into C). Next, using H⋆=A−1PΓ=∑ωγωA−1PEωH =A^-1P_T = _ω _ωA^-1P_TE_ω and the triangle inequality, ‖H⋆‖1≤‖Γ‖1⋅maxω⁡‖A−1PEω‖1. H _1≤ _1· _ω A^-1P_TE_ω _1. For each EωE_ω, ‖A−1PEω‖1≤‖A−1‖1→1‖PEω‖1, A^-1P_TE_ω _1≤ A^-1 _1→ 1\, P_TE_ω _1, and by symmetry of A−1A^-1, ‖A−1‖1→1=‖A−1‖∞→∞ A^-1 _1→ 1= A^-1 _∞→∞. By Assumption C.5, ‖A−1‖∞→∞≤CAσ2d⋆ A^-1 _∞→∞≤ C_A\,σ^2d , and since P_T is symmetric, maxω⁡‖PEω‖1=‖P‖1→1=‖P‖∞→∞=poly(μ,r,m) _ω P_TE_ω _1= P_T _1→ 1= P_T _∞→∞=poly(μ,r,m) (Lemma C.36), which is dimension-free. Hence 1d⋆‖A−1‖∞→∞⋅maxω⁡‖PEω‖1=CAσ2d⋆d⋆⋅poly(μ,r,m)=CAσ2⋅poly(μ,r,m). 1d \; A^-1 _∞→∞· _ω P_TE_ω _1\;=\; C_A\,σ^2d d ·poly(μ,r,m)\;=\;C_A\,σ^2·poly(μ,r,m). The σ2σ^2 cancels the 1/σ21/σ^2 prefactor, yielding (C.53). □ Remark C.21 (ℓ1 _1 dimension cancellation). The key mechanism is ‖A−1PEω‖1/d⋆≤‖A−1‖∞→∞⋅‖P‖∞→∞/d⋆=CAσ2⋅poly(μ,r,m). A^-1P_TE_ω _1/d ≤ A^-1 _∞→∞· P_T _∞→∞/d =C_A\,σ^2·poly(μ,r,m). The d⋆d in ‖A−1‖∞→∞≍σ2d⋆ A^-1 _∞→∞ σ^2d cancels exactly with the 1/d⋆1/d from the comparison averaging, and ‖P‖∞→∞=poly(μ,r,m) P_T _∞→∞=poly(μ,r,m) is dimension-free. This gives |R2|≤C(μ,r,m)CA‖Δ‖∞2‖Γ‖1|R_2|≤ C(μ,r,m)\,C_A\, _∞^2\, _1 without any residual d¯ d factor. C.10 Additional Error from Estimated Information G G As discussed in the proof strategy (Section C.3), the analysis in Sections C.4–C.9 was carried out using the true information operator G=⋆[I(η⋆(X))(X⊗X)]G=E [I(η (X))(X X)], i.e., the estimated direction was defined as H^=(P^GP^)−1P^Γ H=( P_TG P_T)^-1 P_T . In practice, G is replaced by the plug-in estimate G^=⋆[I(η^(X))(X⊗X)] G=E [I( η(X))(X X)], yielding A^′:=P^G^P A := P_T G P_T in place of A^:=P^GP A:= P_TG P_T. We now show that this substitution introduces only a negligible multiplicative correction. Perturbation bound for G^−G G-G. The only difference between G G and G is the use of I(η^(x))I( η(x)) in place of I(η⋆(x))I(η (x)). Since the Fisher information I(⋅)I(·) is Lipschitz on the bounded-signal domain (Assumption C.3), we have |I(η^(x))−I(η⋆(x))|≤LI|η^(x)−η⋆(x)|=LI|⟨Δ,x⟩|≤ 2LI‖Δ‖∞|I( η(x))-I(η (x))|\;≤\;L_I\,| η(x)-η (x)|\;=\;L_I\,| ,x |\;≤\;2L_I\, _∞ for each comparison atom x, where LIL_I is the Lipschitz constant of I on [−B−1,B+1][-B-1,B+1]. Therefore ‖G^−G‖∞→∞≤C‖Δ‖∞σ2d⋆,\| G-G\|_∞→∞\;≤\; C\, _∞σ^2d , where we used ‖G‖∞→∞≍1/(σ2d⋆)\|G\|_∞→∞ 1/(σ^2d ). Define the relative perturbation εG:=‖G^−G‖∞→∞‖G‖∞→∞≤C‖Δ‖∞≤Cσd¯logc⁡d¯n≪ 1. _G\;:=\; \| G-G\|_∞→∞\|G\|_∞→∞\;≤\;C\, _∞\;≤\;C\,σ d ^c dn\; \;1. (C.54) Effect on the restricted inverse. Write A^′=A^+P^(G^−G)P A = A+ P_T( G-G) P_T, so A^′⁣−1=A^−1(I+A^−1P^(G^−G)P^)−1. A -1\;=\; A^-1 (I+ A^-1 P_T( G-G) P_T )^-1. The perturbation term satisfies ‖A^−1P^(G^−G)P^‖∞→∞≤‖A^−1P^G‖∞→∞⋅εG≤C(μ,r,m)⋅εG\| A^-1 P_T( G-G) P_T\|_∞→∞≤\| A^-1 P_TG\|_∞→∞· _G≤ C(μ,r,m)· _G, where we used Remark C.6. By the Neumann series, when εG _G is sufficiently small: A^′⁣−1=A^−1(I+O(εG))in ∥⋅∥∞→∞. A -1= A^-1 (I+O( _G) ) \|·\|_∞→∞. Impact on each error term. The replacement A^→A^′ A→ A affects only the terms involving the estimated direction H H: • RempHR_emp^H and the H-direction bias: the direction error H^′−H⋆ H -H acquires a multiplicative (1+O(εG))(1+O( _G)) correction. Since εG=O(‖Δ‖∞) _G=O( _∞) is of the same order as the existing subspace estimation error, this is absorbed into the combined bound without changing the leading-order rate. • RempηR_emp^η, RprojR_proj, the first-order cancellation, and the second-order remainder all involve only the true direction H⋆H or the true operator G, so they receive no additional error from the G G substitution. In summary, all bounds in this appendix remain valid with G G in place of G, up to a multiplicative (1+O(εG))(1+O( _G)) factor that is negligible under the sample size condition. C.11 Combined Bias Bound We now collect all bias terms from Sections C.6–C.9 and assemble the final bound. C.11.1 Recap of individual bounds The total bias decomposes as Bias=Rproj⏟Sec. C.6+⟨GΔ,H^−H⋆⟩⏟H-bias 1st, Sec. C.7+(H-bias 2nd)⏟Sec. C.7+⟨PΓ,Δ⟩−⟨Δ,G(H⋆)⟩⏟1st cancel, Sec. C.8+R2(Δ;H⋆)⏟2nd order, Sec. C.9.Bias= R_proj_Sec.~ app:pd-Rproj+ G , H-H _H-bias 1st, Sec.~ app:pd-H-bias+ (H-bias 2nd)_Sec.~ app:pd-H-bias+ P_T , - ,G(H ) _1st cancel, Sec.~ app:pd-1st-cancel+ R_2( ;H )_2nd order, Sec.~ app:pd-2nd-order. (C.55) The individual bounds are: (i) Projection leakage (Eq. (C.32)): |Rproj|≤‖Γ‖1⋅‖(I−P)Δ‖∞|R_proj|≤ _1· (I-P_T) _∞. (i) H-direction bias, first order (Prop. C.16): |⟨GΔ,H^−H⋆⟩|≤C(μ,r,m)CA‖Γ‖1[‖(I−P^)Δ‖∞+‖(I−P)Δ‖∞].| G , H-H |≤ C(μ,r,m)\,C_A\, _1 [ (I- P_T) _∞+ (I-P_T) _∞ ]. (i) First-order cancellation (Prop. C.18): |⟨PΓ,Δ⟩−⟨Δ,G(H⋆)⟩|≤C(μ,r,m)CA‖Γ‖1‖(I−P)Δ‖∞.| P_T , - ,G(H ) |≤ C(μ,r,m)\,C_A\, _1\, (I-P_T) _∞. (iv) Second-order remainder (Prop. C.20): |R2(Δ;H⋆)|≤C(μ,r,m)CA‖Δ‖∞2‖Γ‖1|R_2( ;H )|≤ C(μ,r,m)\,C_A\, _∞^2\, _1. (v) H-direction bias, second order (Eq. (C.46)): controlled by CCA‖Δ‖∞2‖Γ‖1C\,C_A\, _∞^2\, _1; same order as (iv). C.11.2 Combined bound Theorem C.22 (Combined bias bound). Under Assumptions C.3 and C.5, the total bias satisfies |Bias|≤C(μ,r,m)CA‖Γ‖1[‖(I−P)Δ‖∞+‖(I−P^)Δ‖∞+‖Δ‖∞2].|Bias|\;≤\;C(μ,r,m)\,C_A\, _1 [ (I-P_T) _∞+ (I- P_T) _∞+ _∞^2 ]. (C.56) Proof of Theorem C.22. Sum the bounds (i)–(iv): terms (i) and (i) contribute C‖Γ‖1‖(I−P)Δ‖∞C\, _1\, (I-P_T) _∞; term (i) contributes C‖Γ‖1[‖(I−P^)Δ‖∞+‖(I−P)Δ‖∞]C\, _1[ (I- P_T) _∞+ (I-P_T) _∞]; term (iv) contributes C‖Γ‖1‖Δ‖∞2C\, _1\, _∞^2. Collecting with the maximum constant gives (C.56). □ C.11.3 Substituting the ε2 ^2 and entrywise rates We now plug in the standard tensor estimation rates to obtain a fully explicit final bound. Input rates. Recall from Section C.6.3: • Normal-component error (Proposition C.13): ‖(I−P)Δ‖∞,‖(I−P^)Δ‖∞≤C(μ,r,m)ρ2 (I-P_T) _∞,\; (I- P_T) _∞\;≤\;C(μ,r,m)\,ρ^2, where ρ:=σλmind⋆d¯log⁡d¯nρ:= σ _ d \, d dn. • Entrywise error: ‖Δ‖∞≤C(μ,r,m)⋅σd¯/n _∞≤ C(μ,r,m)·σ d/n. Substitution into each term. (i) First-order cancellation (Prop. C.18): CCA‖Γ‖1‖(I−P)Δ‖∞≤C(μ,r,m)CA‖Γ‖1ρ2=C(μ,r,m)CA‖Γ‖1⋅σ2d⋆d¯log⁡d¯λmin2n.C\,C_A\, _1\, (I-P_T) _∞\;≤\;C(μ,r,m)\,C_A\, _1\,ρ^2\;=\;C(μ,r,m)\,C_A\, _1· σ^2\,d \, d d _ ^2\,n. (i) Second-order remainder (Prop. C.20): CCA‖Γ‖1‖Δ‖∞2≤C(μ,r,m)CA‖Γ‖1⋅σ2d¯n.C\,C_A\, _1\, _∞^2\;≤\;C(μ,r,m)\,C_A\, _1· σ^2\, dn. (i) Projection leakage (Eq. (C.38)): |Rproj|≤C(μ,r,m)‖Γ‖1ρ2=C(μ,r,m)‖Γ‖1⋅σ2d⋆d¯log⁡d¯λmin2n.|R_proj|\;≤\;C(μ,r,m)\, _1\,ρ^2\;=\;C(μ,r,m)\, _1· σ^2\,d \, d d _ ^2\,n. (iv) H-direction bias (Prop. C.16): controlled by CCA‖Γ‖1[‖(I−P)Δ‖∞+‖(I−P^)Δ‖∞+‖Δ‖∞2]≤CCA‖Γ‖1[ρ2+‖Δ‖∞2].C\,C_A\, _1 [ (I-P_T) _∞+ (I- P_T) _∞+ _∞^2 ]≤ C\,C_A\, _1 [ρ^2+ _∞^2 ]. Final explicit bound. Under the SNR condition σ/λmin⋅d⋆≍1σ/ _ · d 1, the ρ2ρ^2 and ‖Δ‖∞2 _∞^2 terms are the same order; both are retained. Combining: |Bias|≤C(μ,r,m)CA∥Γ∥1[σ2d⋆d¯log⁡d¯λmin2n+σ2d¯n]. |Bias|\;≤\;C(μ,r,m)\,C_A\, _1 [ σ^2\,d \, d d _ ^2\,n+ σ^2\, dn ]. (C.57) Remark C.23 (Pairwise-comparison specialization). In the pairwise-comparison setting, σ=O(1)σ=O(1) and λmin≍d⋆ _ d (Appendix C.1). All bounds simplify because the two key estimation parameters collapse to the same scale: ρ=σλmind⋆d¯log⁡d¯n≍d¯logc⁡d¯n,δ∞:=‖Δ‖∞≲d¯logc⁡d¯n.ρ\;=\; σ _ d d dn\; \; d ^c dn, _∞\;:=\; _∞\; \; d ^c dn. (C.58) In particular, both ρ and δ∞ _∞ are of order d¯logc⁡d¯/n d ^c d/n, and the products ρ2ρ^2, δ∞2 _∞^2, ρ⋅δ∞ρ· _∞ are all of order d¯logc⁡d¯/n d ^c d/n. Substituting these rates into the combined bounds (C.57) and (C.27), with δ=d¯−cδ= d^-c (so log⁡(2/δ)≍clog⁡d¯ (2/δ) c d), the total remainder satisfies |Rn|≤C(μ,r,m)CA‖Γ‖1d¯logc⁡d¯n |R_n|\;≤\;C(μ,r,m)\,C_A\, _1\, d ^c dn (C.59) with probability ≥1−d¯−c≥ 1- d^-c. This is the bound stated in Theorem 4.4 of the main text. Each of the six individual terms (RempHR_ emp^H, RempηR_ emp^η, RprojR_ proj, RbiasHR_ bias^H, first-order cancellation, second-order remainder) contributes at most this order, as summarized in Table 1. C.12 Range of CAC_A and dimension-free conditions This subsection proves Proposition 4.6: the coarse range C(μ,r,m)≤CA≤C(μ,r,m)d¯C(μ,r,m)≤ C_A≤ C(μ,r,m) d, and the dimension-free regime under bounded signal. Throughout, we work under the bounded-signal assumption ‖T⋆‖∞≤B\|T \|_∞≤ B, so the Fisher information satisfies cB≤I(η⋆(X))≤CBc_B≤ I(η (X))≤ C_B for constants depending only on B. Recall that A−1A^-1 acts only on the tangent space T; accordingly, we define ‖A−1‖∞→∞:=sup‖V‖∞≤1‖A−1PV‖∞. A^-1 _∞→∞:= _\|V\|_∞≤ 1\|A^-1P_TV\|_∞. C.12.1 Spectral scale of the restricted inverse Lemma C.24 (Spectral scale of A). Under bounded signal, cBd⋆P⪯A⪯CBd⋆Pon . c_Bd \,P_T A C_Bd \,P_T T. Consequently, ‖A−1‖op≍d⋆\|A^-1\|_op d , with constants depending only on B. Proof. For any H∈H , ⟨H,AH⟩=⋆[I(η⋆(X))⟨H,X⟩2] H,AH =E_ [I(η (X)) H,X ^2]. The Fisher bounds give cB⋆⟨H,X⟩2≤⟨H,AH⟩≤CB⋆⟨H,X⟩2.c_B\,E_ H,X ^2≤ H,AH ≤ C_B\,E_ H,X ^2. Under the uniform pairwise-comparison design, the Frobenius reduction (Lemma C.30) gives ⋆⟨H,X⟩2≍‖H‖F2/d⋆E_ H,X ^2 \|H\|_F^2/d , and the claim follows. ∎ C.12.2 Entrywise bound for tangent-space elements The upper bound CA≲d¯C_A d is not a generic ambient-space norm inequality (which would only yield d⋆ d ). The sharper factor comes from the low-rank tangent-space geometry. Lemma C.25 (Tangent-space entrywise bound). Assume μ-incoherence. For every W∈W , ‖W‖∞≤C(μ,r,m)d¯d⋆‖W‖F.\|W\|_∞≤ C(μ,r,m) dd \,\|W\|_F. Proof. Write the Tucker tangent-space decomposition =Π∅(ℋ)⊕⨁j=1mΠj(ℋ)T= _ (H) _j=1^m _\j\(H), where Π∅ _ is the fully projected block and Πj _\j\ is the single-mode correction block. Any W∈W decomposes orthogonally as W=W∅+∑j=1mWjW=W_ + _j=1^mW_j. For the main block W∅W_ , every element lies in the span of tensors a(1)⊗⋯⊗a(m)a^(1) ·s a^(m) with a(k)∈col(Uk)a^(k) (U_k). By incoherence, ‖a(k)‖∞≤μrk/dk‖a(k)‖2\|a^(k)\|_∞≤ μ r_k/d_k\,\|a^(k)\|_2, so for a rank-one tensor in this block, ‖a(1)⊗⋯⊗a(m)‖∞≤∏k=1mμrk/dk∏k=1m‖a(k)‖2=C(μ,r,m)(d⋆)−1/2‖a(1)⊗⋯⊗a(m)‖F.\|a^(1) ·s a^(m)\|_∞≤ _k=1^m μ r_k/d_k\; _k=1^m\|a^(k)\|_2=C(μ,r,m)\,(d )^-1/2\,\|a^(1) ·s a^(m)\|_F. By linearity, ‖W∅‖∞≤C(μ,r,m)(d⋆)−1/2‖W∅‖F\|W_ \|_∞≤ C(μ,r,m)\,(d )^-1/2\,\|W_ \|_F. For a single-mode block WjW_j, the range is generated by tensors with one free mode and all remaining modes projected. The free mode provides no incoherence gain (‖b(j)‖∞≤‖b(j)‖2\|b^(j)\|_∞≤\|b^(j)\|_2), while each projected mode k≠jk≠ j still satisfies ‖a(k)‖∞≤μrk/dk‖a(k)‖2\|a^(k)\|_∞≤ μ r_k/d_k\,\|a^(k)\|_2. Therefore ‖Wj‖∞≤C(μ,r,m)(∏k≠jdk−1/2)‖Wj‖F=C(μ,r,m)djd⋆‖Wj‖F≤C(μ,r,m)d¯d⋆‖Wj‖F.\|W_j\|_∞≤ C(μ,r,m) ( _k≠ jd_k^-1/2 )\|W_j\|_F=C(μ,r,m) d_jd \,\|W_j\|_F≤ C(μ,r,m) dd \,\|W_j\|_F. Combining via the triangle inequality and the orthogonality of the m+1m+1 blocks (m is fixed) gives the claim. ∎ Remark C.26. This lemma is the precise source of the factor d¯ d: the free mode in each single-mode correction block introduces a dj d_j factor that cannot be controlled by incoherence. C.12.3 Coarse range of CAC_A Proposition C.27 (Coarse range of CAC_A). Under bounded signal and μ-incoherence, c(μ,r,m)≤CA≤C(μ,r,m)d¯.c(μ,r,m)≤ C_A≤ C(μ,r,m) d. Proof. Lower bound. Since ‖M‖∞→∞≥‖M‖op M _∞→∞≥\|M\|_op for any linear operator, Lemma C.24 gives ‖A−1‖∞→∞≥‖A−1‖op≥c(B)d⋆, A^-1 _∞→∞≥\|A^-1\|_op≥ c(B)\,d , so CA=‖A−1‖∞→∞/d⋆≥c(B)C_A= A^-1 _∞→∞/d ≥ c(B). Upper bound. Let V satisfy ‖V‖∞≤1\|V\|_∞≤ 1 and set W:=A−1PV∈W:=A^-1P_TV . By Lemma C.25, ‖W‖∞≤C(μ,r,m)d¯d⋆‖W‖F.\|W\|_∞≤ C(μ,r,m) dd \,\|W\|_F. Since ‖W‖F≤‖A−1‖op‖PV‖F≤C(B)d⋆⋅d⋆\|W\|_F≤\|A^-1\|_op\|P_TV\|_F≤ C(B)\,d · d (using ‖V‖F≤d⋆‖V‖∞\|V\|_F≤ d \,\|V\|_∞), we obtain ‖A−1‖∞→∞≤C(μ,r,m)d¯d⋆⋅C(B)d⋆d⋆=C(μ,r,m,B)d⋆d¯, A^-1 _∞→∞≤ C(μ,r,m) dd · C(B)\,d d =C(μ,r,m,B)\,d d, giving CA≤C(μ,r,m,B)d¯C_A≤ C(μ,r,m,B) d. ∎ C.12.4 Dimension-free CAC_A under bounded signal When the Fisher weights are nearly constant, a perturbative argument around the isotropic benchmark shows that CAC_A is dimension-free. Proposition C.28 (Dimension-free CAC_A in the near-constant regime). If ‖T⋆‖∞≤B0\|T \|_∞≤ B_0 where B0=B0(μ,m,r)B_0=B_0(μ,m,r) is sufficiently small depending only on the structural parameters, then ‖A−1‖∞→∞∨‖A^−1‖∞→∞≤C(μ,r,m,B0)d⋆. A^-1 _∞→∞ A^-1 _∞→∞≤ C(μ,r,m,B_0)\,d . In particular, CA≤C(μ,r,m,B0)C_A≤ C(μ,r,m,B_0), a dimension-free constant. Proof. Constant-weight benchmark. Let I0:=I(0)=1/4I_0:=I(0)=1/4 and define the benchmark operator A0:=PG0P,G0(U):=⋆[I0⟨U,X⟩X].A_0:=P_TG_0P_T, G_0(U):=E_ [I_0 U,X X]. Under constant Fisher weights, the information operator is isotropic on the identifiable comparison space: A0=(I0/cpwd⋆)PA_0=(I_0/c_pwd )\,P_T, where cpwc_pw is the pairwise-design normalization constant. Hence A0−1=cpwd⋆I0P,‖A0−1‖∞→∞=cpwd⋆I0‖P‖∞→∞≤C(μ,r,m)d⋆,A_0^-1= c_pwd I_0\,P_T, A_0^-1 _∞→∞= c_pwd I_0 P_T _∞→∞≤ C(μ,r,m)\,d , where the last step uses ‖P‖∞→∞=poly(μ,r,m) P_T _∞→∞=poly(μ,r,m) (Lemma C.36). Fisher-weight perturbation. Since I(η)I(η) is smooth and even, for |η|≤2B0|η|≤ 2B_0 we have |I(η)−I0|≤LIB0|I(η)-I_0|≤ L_IB_0 for an absolute constant LIL_I. Write I(η⋆(X))=I0+δ(X),‖δ‖∞≤ε(B0)→0 as B0→0.I(η (X))=I_0+δ(X), \|δ\|_∞≤ (B_0)→ 0 as B_0→ 0. Decompose G=G0+EG=G_0+E and A=A0+ΔA=A_0+ _A with ΔA:=PEP _A:=P_TEP_T. For the pairwise design, each design tensor X has two nonzero entries equal to ±1± 1, so |⟨U,X⟩|≤2‖U‖∞| U,X |≤ 2\|U\|_∞. Since the coordinate ω is involved with probability O(1/d⋆)O(1/d ), ‖E‖∞→∞≤Cε(B0)(d⋆)−1,‖ΔA‖∞→∞≤‖P‖∞→∞2‖E‖∞→∞≤C(μ,r,m)ε(B0)(d⋆)−1. E _∞→∞≤ C\, (B_0)\,(d )^-1, _A _∞→∞≤ P_T _∞→∞^2 E _∞→∞≤ C(μ,r,m)\, (B_0)\,(d )^-1. Neumann expansion. The normalized perturbation satisfies ρ:=‖A0−1ΔA‖∞→∞≤‖A0−1‖∞→∞‖ΔA‖∞→∞≤C(μ,r,m)ε(B0).ρ:= A_0^-1 _A _∞→∞≤ A_0^-1 _∞→∞ _A _∞→∞≤ C(μ,r,m)\, (B_0). Choose B0B_0 small enough that ρ≤1/2ρ≤ 1/2. Then A=A0(I+A0−1ΔA)A=A_0(I+A_0^-1 _A) with ‖(I+A0−1ΔA)−1‖∞→∞≤∑k=0∞ρk=11−ρ≤2,\|(I+A_0^-1 _A)^-1\|_∞→∞≤ _k=0^∞ρ^k= 11-ρ≤ 2, so ‖A−1‖∞→∞≤2‖A0−1‖∞→∞≤C(μ,r,m,B0)d⋆ A^-1 _∞→∞≤ 2 A_0^-1 _∞→∞≤ C(μ,r,m,B_0)\,d . Estimated operator. The same argument applies to A^=P^G^P A= P_T G P_T on the high-probability event that ‖P^‖∞→∞≤C(μ,r,m),‖G^−G‖∞→∞=o((d⋆)−1). P_T _∞→∞≤ C(μ,r,m), G-G _∞→∞=o((d )^-1). ∎ C.13 Relative error of plug-in variance estimators This subsection proves the variance estimation bounds (4.14) and (5.9). C.13.1 Algebraic reduction to relative direction error The following lemma applies to both the efficient and whitened settings. Lemma C.29 (Variance difference reduces to direction error). Let ϕ^(X,Y)=s⋆(Y,X)⟨H^,X⟩ φ(X,Y)=s_ (Y,X) H,X and ϕ⋆(X,Y)=s⋆(Y,X)⟨H⋆,X⟩φ (X,Y)=s_ (Y,X) H ,X , where s⋆s_ denotes either the usual score sη(Y,η⋆)s_η(Y,η ) or the whitened score s~(Y,η⋆) s(Y,η ). Assume the score-squared factor satisfies ⋆[s⋆2f(X)]≍⋆[f(X)]E [s_ ^2f(X)] [f(X)] for nonnegative f. If the relative direction error δH:=‖H^−H⋆‖F‖H⋆‖F _H:= \| H-H \|_F\|H \|_F is sufficiently small, then the direction-induced relative variance error satisfies |ℙ⋆[s⋆2(⟨H^,X⟩2−⟨H⋆,X⟩2)]|ℙ⋆[s⋆2⟨H⋆,X⟩2]≲δH. |P [s_ ^2( H,X ^2- H ,X ^2) ] |P [s_ ^2 H ,X ^2]\; \; _H. Proof. Write a:=⟨H^,X⟩a:= H,X and b:=⟨H⋆,X⟩b:= H ,X . By score comparability, |ℙ⋆[s⋆2(a2−b2)]|≲ℙ⋆|a2−b2||P [s_ ^2(a^2-b^2)]| |a^2-b^2|. The factorization a2−b2=(a−b)(a+b)a^2-b^2=(a-b)(a+b) and Cauchy–Schwarz give ℙ⋆|a2−b2|≤(ℙ⋆(a−b)2)1/2(ℙ⋆(a+b)2)1/2.P |a^2-b^2|≤ (P (a-b)^2 )^1/2 (P (a+b)^2 )^1/2. By the pairwise Frobenius reduction (Lemma C.30), ℙ⋆(a−b)2≍‖H^−H⋆‖F2/d⋆,ℙ⋆b2≍‖H⋆‖F2/d⋆.P (a-b)^2 \| H-H \|_F^2/d , b^2 \|H \|_F^2/d . When δH≤c0 _H≤ c_0, (a+b)2≤2(a−b)2+8b2(a+b)^2≤ 2(a-b)^2+8b^2 gives ℙ⋆(a+b)2≤10ℙ⋆b2P (a+b)^2≤ 10\,P b^2. Substituting and dividing by ℙ⋆[s⋆2b2]≍ℙ⋆b2P [s_ ^2b^2] b^2 yields the claim. ∎ C.13.2 Efficient estimator: proof of (4.14) The total relative variance error decomposes into three parts: empirical fluctuation, score plug-in, and direction-induced. Direction-induced part. The efficient directions satisfy H⋆=A−1PΓH =A^-1P_T and H^=A^−1P^Γ H= A^-1 P_T . Add and subtract A−1P^ΓA^-1 P_T and use the resolvent identity A^−1−A−1=A−1(A−A^)A^−1 A^-1-A^-1=A^-1(A- A) A^-1: H^−H⋆=A−1(P^−P)Γ⏟T1+A−1(A−A^)A^−1P^Γ⏟T2. H-H = A^-1( P_T-P_T) _T_1+ A^-1(A- A) A^-1 P_T _T_2. Since A has spectral scale (d⋆)−1(d )^-1 on the tangent space, A−1A^-1 multiplies Frobenius norms by ≍d⋆ d . This common factor cancels in the relative ratio ‖T1‖F/‖H⋆‖F\|T_1\|_F/\|H \|_F, yielding ‖T1‖F‖H⋆‖F≲‖(P^−P)Γ‖F‖PΓ‖F. \|T_1\|_F\|H \|_F \|( P_T-P_T) \|_F\|P_T \|_F. Under the sparse-target projector perturbation bound ‖(P^−P)Γ‖F≲ρd¯/d⋆‖Γ‖F\|( P_T-P_T) \|_F ρ d/d \,\| \|_F and the alignment condition ‖PΓ‖F≥αΓd¯/d⋆‖Γ‖F\|P_T \|_F≥ _ d/d \,\| \|_F, this is O(ρ/αΓ)O(ρ/ _ ). For T2T_2, the operator perturbation ‖A−A^‖op≲ρ/(d⋆)\|A- A\|_op ρ/(d ) together with ‖A−1‖op≍d⋆\|A^-1\|_op d gives ‖A−1(A−A^)‖op≲ρ\|A^-1(A- A)\|_op ρ, so ‖T2‖F≲ρ‖H^‖F≍ρ‖H⋆‖F\|T_2\|_F ρ\| H\|_F ρ\|H \|_F. Combining, δH:=‖H^−H⋆‖F‖H⋆‖F≲ραΓ. _H:= \| H-H \|_F\|H \|_F ρ _ . Lemma C.29 then gives a direction-induced relative error of O(ρ/αΓ)O(ρ/ _ ). Score plug-in part. By the derivative bound on the score, |sη(Y,η^)2−sη(Y,η⋆)2|≲‖T^−T⋆‖∞|s_η(Y, η)^2-s_η(Y,η )^2| \| T-T \|_∞, so the relative contribution is O(‖T^−T⋆‖∞)O(\| T-T \|_∞). Empirical fluctuation. Conditional on the first-stage sample, V^eff−ℙ⋆[ϕ^2]=(ℙn−ℙ⋆)(ϕ^2) V_ eff-P [ φ^2]=(P_n-P )( φ^2) is a mean-zero average. A Bernstein bound gives relative error O(d¯log⁡d¯/n)O( d d/n). Combining the three parts yields (4.14). C.13.3 Score-whitened estimator: proof of (5.9) The argument simplifies because the whitened directions Hws⋆=d⋆PΓH_ ws =d P_T and H^ws=d⋆P^Γ H_ ws=d P_T involve no operator inversion. The direction difference is H^ws−Hws⋆=d⋆(P^−P)Γ, H_ ws-H_ ws =d ( P_T-P_T) , so the d⋆d factor cancels in the relative ratio: ‖H^ws−Hws⋆‖F‖Hws⋆‖F=‖(P^−P)Γ‖F‖PΓ‖F≲ραΓ. \| H_ ws-H_ ws \|_F\|H_ ws \|_F= \|( P_T-P_T) \|_F\|P_T \|_F ρ _ . Lemma C.29 (applied with the whitened score s~ s) gives the direction-induced relative error. The score plug-in and empirical fluctuation parts are identical to the efficient case, completing the proof of (5.9). C.14 Useful Technical Tools C.14.1 Frobenius reduction for comparison sampling Lemma C.30 (Pairwise difference identity). Fix any vector z∈ℝd1z ^d_1 with ∑a=1d1za=0 _a=1^d_1z_a=0. If a,b\a,b\ is uniformly distributed over unordered pairs, then a,b[(za−zb)2]=2d1−1‖z‖22.E_\a,b\ [(z_a-z_b)^2 ]= 2d_1-1\, z _2^2. (C.60) Proof of Lemma C.30. We have ∑a<b(za−zb)2=d1∑aza2−(∑aza)2=d1‖z‖22 _a<b(z_a-z_b)^2=d_1 _az_a^2-( _az_a)^2=d_1 z _2^2. Dividing by (d12) d_12 yields (C.60). □ Lemma C.31 (Frobenius reduction for comparison sampling). Let H satisfy the column-sum-zero condition (C.5). Under the comparison sampling model, ⋆[⟨H,X⟩2]=2D(d1−1)‖H‖F2≍‖H‖F2d⋆.E [ H,X ^2 ]= 2D(d_1-1)\, H _F^2 H _F^2d . (C.61) Proof of Lemma C.31. Write hu∈ℝd1h_u ^d_1 for the u-th context slice. Conditioning on u and applying Lemma C.30: ⋆[⟨H,X⟩2]=1D∑u2d1−1‖hu‖22=2D(d1−1)‖H‖F2E [ H,X ^2]= 1D _u 2d_1-1 h_u _2^2= 2D(d_1-1) H _F^2. □ Corollary C.32 (Weighted second moment). Under Assumption C.3 and Lemma C.31, for any column-sum-zero H, ⋆[I(η⋆(X))⟨H,X⟩2]≤c+σ2⋅‖H‖F2d⋆.E \! [I(η (X))\, H,X ^2 ]≤ c_+σ^2· H _F^2d . (C.62) C.14.2 ℓ1 _1 extraction Lemma C.33 (ℓ1 _1 extraction). For any linear map ℳM and any Γ=∑ωγωeω = _ω _ωe_ω, ‖ℳΓ‖F≤‖Γ‖1maxω⁡‖ℳeω‖F, \, _F≤ _1\ _ω\ _ω _F, (C.63) ‖ℳΓ‖∞≤‖Γ‖1maxω⁡‖ℳeω‖∞, \, _∞≤ _1\ _ω\ _ω _∞, ‖ℳΓ‖1≤‖Γ‖1maxω⁡‖ℳeω‖1. \, _1≤ _1\ _ω\ _ω _1. Proof of Lemma C.33. By linearity and the triangle inequality, ‖ℳΓ‖F≤∑ω|γω|‖ℳeω‖F≤‖Γ‖1maxω⁡‖ℳeω‖F _F≤ _ω| _ω|\, _ω _F≤ _1\, _ω _ω _F. The ℓ∞ _∞ and ℓ1 _1 versions are identical. □ C.14.3 Block projections and tangent space for Tucker structure Let ℋ:=ℝd1×⋯×dmH:=R^d_1×·s× d_m with Frobenius inner product. Fix orthonormal matrices Uj∈ℝdj×rjU_j ^d_j× r_j and define Pj:=UjUj⊤P_j:=U_jU_j , Pj⟂:=Idj−PjP_j :=I_d_j-P_j. For any subset S⊆[m]S [m], define the block projector ΠS(X):=X×j∈SPj⟂×k∉SPk. _S(X):=X×_j∈ SP_j ×_k∉ SP_k. (C.64) Lemma C.34 (Orthogonal block decomposition). The family ΠSS⊆[m]\ _S\_S [m] consists of orthogonal projections with pairwise orthogonal ranges and ∑S⊆[m]ΠS=I _S [m] _S=I. Proof of Lemma C.34. Each Pj,Pj⟂P_j,P_j is symmetric idempotent, and products in distinct modes commute. If S≠S′S≠ S , pick j∈S∖S′j∈ S S ; then ΠS _S multiplies by Pj⟂P_j and ΠS′ _S by PjP_j, giving ΠSΠS′=0 _S _S =0. For the resolution, ∏j(Pj+Pj⟂)=I _j(P_j+P_j )=I, and expanding yields ∑SΠS _S _S. □ Lemma C.35 (Tangent space is “|S|≤1|S|≤ 1” blocks). Under full multilinear rank, =Π∅(ℋ)⊕⨁j=1mΠj(ℋ),⟂=⨁S⊆[m]|S|≥2ΠS(ℋ).T= _ (H)\; \; _j=1^m _\j\(H), = _ subarraycS [m]\\ |S|≥ 2 subarray _S(H). (C.65) In particular, (I−P)Δ=∑|S|≥2ΠS(Δ)(I-P_T) = _|S|≥ 2 _S( ). Proof of Lemma C.35. For the inclusion ⊆ : each term in the tangent space representation δC×1U1⋯×mUmδ C×_1U_1·s×_mU_m lies in Π∅(ℋ) _ (H), and each C⋆×jWj×k≠jUkC ×_jW_j×_k≠ jU_k (with Uj⊤Wj=0U_j W_j=0) lies in Πj(ℋ) _\j\(H). For the inclusion ⊇ : any Y∈Π∅(ℋ)Y∈ _ (H) can be written as C~×1U1⋯×mUm∈ C×_1U_1·s×_mU_m , and any Y∈Πj(ℋ)Y∈ _\j\(H) has mode-j fibers in Range(Pj⟂)Range(P_j ); by full rank of C(j)⋆C _(j), one can find WjW_j with Uj⊤Wj=0U_j W_j=0 such that Y=C⋆×jWj×k≠jUk∈Y=C ×_jW_j×_k≠ jU_k . The orthogonal complement formula follows from Lemma C.34. □ C.14.4 Dimension-free ∞→∞\!→\!∞ bounds for projections Lemma C.36 (Dimension-free bounds). Let P=UU⊤P=U with maxi⁡‖ei⊤U‖22≤μr/d _i\|e_i U\|_2^2≤μ r/d. Then ‖P‖∞→∞≤μr,‖P⟂‖∞→∞≤1+μr. P _∞→∞≤ μ\,r, P _∞→∞≤ 1+ μ\,r. (C.66) For the centering Q=I−1d⊤Q=I- 1d11 : ‖Q‖∞→∞<2 Q _∞→∞<2 and ‖Q−P‖∞→∞≤2+μr Q-P _∞→∞≤ 2+ μ\,r. For Kronecker products: ‖⨂jAj‖∞→∞=∏j‖Aj‖∞→∞ _jA_j _∞→∞= _j A_j _∞→∞. For the tangent projector: ‖P‖∞→∞≤∑|S|≤1‖ΠS‖∞→∞=poly(μ,r,m) P_T _∞→∞≤ _|S|≤ 1 _S _∞→∞=poly(μ,r,m). Proof of Lemma C.36. Row i of P has entries Pij=⟨ui,uj⟩P_ij= u_i,u_j , so ∑j|Pij|≤‖ui‖2dr=μr/d⋅dr=μr _j|P_ij|≤\|u_i\|_2 d r= μ r/d· dr= μ\,r. For P⟂P , ‖I−P‖∞→∞≤1+‖P‖∞→∞ I-P _∞→∞≤ 1+ P _∞→∞. For Q, each row has sum |1−1/d|+(d−1)/d=2−2/d<2|1-1/d|+(d-1)/d=2-2/d<2. For Kronecker products, absolute row sums multiply. C.14.5 ‖PEω‖∞\|P_TE_ω\|_∞ bound Lemma C.37 (‖PEω‖∞\|P_TE_ω\|_∞ is d/d⋆d/d up to incoherence). Under μ-incoherence, for any canonical basis tensor Eω=ei1⊗⋯⊗eimE_ω=e_i_1 ·s e_i_m, ‖PEω‖∞≤Cμd⋆,Cμ:=(m+1)μm, P_TE_ω _∞≤ C_μ\, dd , C_μ:=(m+1)μ^m, (C.67) where d:=∏jrj(1+∑kdk/rk)d:= _jr_j(1+ _kd_k/r_k) is the effective tangent dimension proxy. Proof of Lemma C.37. Using P=Π∅+∑kΠkP_T= _ + _k _\k\, we have ‖Π∅(Eω)‖∞=∏j‖Pjeij‖∞≤μm∏jrj/dj=μmr⋆/d⋆, _ (E_ω) _∞= _j P_je_i_j _∞≤μ^m _jr_j/d_j=μ^mr /d , and ‖Πk(Eω)‖∞≤‖Pk⟂eik‖∞∏j≠k‖Pjeij‖∞≤μm−1(r⋆/d⋆)(dk/rk). _\k\(E_ω) _∞≤ P_k e_i_k _∞ _j≠ k P_je_i_j _∞≤μ^m-1(r /d )(d_k/r_k). Summing gives ‖PEω‖∞≤μmr⋆/d⋆+∑kμm−1r⋆dk/(rkd⋆)≤(m+1)μm⋅d/d⋆. P_TE_ω _∞≤μ^mr /d + _kμ^m-1r d_k/(r_kd )≤(m+1)μ^m· d/d . □ Sharp ‖Pω‖∞\|P_Tω\|_∞ for comparison design. For the comparison design, ω=eu,a−eu,bω=e_u,a-e_u,b and the Ma-type tangent projector decomposition P=×+∑jℛjP_T=P_×+ _jR_j gives, via the rank-one unfolding argument: ‖Pω‖∞≤poly(μ,r,m)⋅dmaxd⋆.\|P_Tω\|_∞ (μ,r,m)· d_ d . (C.68) The main term ×(ω)P_×(ω) factorizes as ⨂kPUkeik _kP_U_ke_i_k, giving μmr⋆/d⋆μ^mr /d . Each correction ℛj(ω)R_j(ω) is rank-one in mode-j matricization with ‖ℛj(ω)‖∞≤(1+μrj)μm−1r⋆dj/(rjd⋆)\|R_j(ω)\|_∞≤(1+μ r_j)μ^m-1r d_j/(r_jd ). The additional row-sum projection Πk _k satisfies ‖Πk(Z)‖∞≤2‖Z‖∞\| _k(Z)\|_∞≤ 2\|Z\|_∞ and does not affect the d⋆d cancellation. C.14.6 Symmetry and norm equivalences Lemma C.38 (Symmetry: ‖M‖1→1=‖M‖∞→∞\|M\|_1→ 1=\|M\|_∞→∞). For any M∈ℝp×pM ^p× p, ‖M‖1→1=‖M⊤‖∞→∞\|M\|_1→ 1=\|M \|_∞→∞. If M=M⊤M=M , then ‖M‖1→1=‖M‖∞→∞\|M\|_1→ 1=\|M\|_∞→∞. C.14.7 Bernstein template Lemma C.39 (Bernstein for bounded variables). Let W1,…,WnW_1,…,W_n be i.i.d. mean-zero with |Wi|≤M|W_i|≤ M a.s. and Var(Wi)≤vVar(W_i)≤ v. Then for any δ∈(0,1)δ∈(0,1), with probability at least 1−δ1-δ, |1n∑i=1nWi|≤C(vlog⁡(2/δ)n+Mlog⁡(2/δ)n). | 1n _i=1^nW_i |≤ C ( v (2/δ)n+ M (2/δ)n ). C.14.8 Entrywise contraction Lemma C.40 (Entrywise contraction via ∥⋅∥2,∞\|·\|_2,∞). Let G∈ℝr1×⋯×rmG ^r_1×·s× r_m and Aj∈ℝdj×rjA_j ^d_j× r_j. Then ‖G×1A1⋯×mAm‖∞≤‖G‖op∏j‖Aj‖2,∞ G×_1A_1·s×_mA_m _∞≤ G _op _j A_j _2,∞. Proof of Lemma C.40. For each entry (i1,…,im)(i_1,…,i_m), the value is ⟨G,a1(i1)⊗⋯⊗am(im)⟩ G,a_1(i_1) ·s a_m(i_m) where aj(ij)⊤=eij⊤Aja_j(i_j) =e_i_j A_j. By definition of ‖G‖op G _op, |Yi1,…,im|≤‖G‖op∏j‖aj(ij)‖2≤‖G‖op∏j‖Aj‖2,∞.|Y_i_1,…,i_m|≤ G _op _j\|a_j(i_j)\|_2≤ G _op _j A_j _2,∞. □ Appendix D Proofs for Section 5 This appendix proves Theorems 5.1 and 5.2. The proof follows the same five-term decomposition as Appendix C, but each term simplifies dramatically because score whitening turns the information operator into (1/d⋆)I(1/d )I, eliminating the operator inverse and CAC_A entirely. Proof roadmap and comparison with the general case. For ease of reference, we list the five remainder terms side-by-side: Term General (Sec. C) Whitened (this section) Direction error RempHR_ emp^H CAσd¯ρlog/nC_Aσ d\,ρ /n d¯ρlog/n d\,ρ /n Score perturbation RempηR_ emp^η CAδ∞d¯/nC_A\, _∞ d/n δ∞d¯/n _∞ d/n Projection leakage RprojR_ proj ρ2ρ^2 ρ2ρ^2 (unchanged) H-bias (1st order) CAρ2C_A\,ρ^2 ρ2ρ^2 1st-order cancel + 2nd remainder CAρ2+CAδ∞2C_A\,ρ^2+C_A\, _∞^2 0+δ∞20+ _∞^2 The key simplifications are: (i) CAC_A disappears from all terms because A0−1=d⋆PA_0^-1=d P_T is explicit; (i) the σ factors from ‖A−1‖\|A^-1\| and 1/σ1/σ from the score cancel automatically since the whitened score has O(1)O(1) bounds; (i) the first-order cancellation becomes complete (zero, not just O(ρ2)O(ρ^2)), because G0(Hws⋆)=PΓ∈G_0(H _ ws)=P_T has no off-tangent component. D.1 Setup and notation We adopt the comparison sampling model and column-sum-zero restriction of Section C.2. Throughout, s~(y,η):=s(y,η)/I(η) s(y,η):=s(y,η)/I(η) denotes the whitened score (Eq. (5.1)). Its key properties, inherited from the exponential family structure, are: (i) Centering: ⋆[s~(Y,η⋆)∣X]=0E [ s(Y,η ) X]=0. (i) Constant conditional derivative: ⋆[∂ηs~(Y,η⋆)∣X]=−1E [ _η s(Y,η ) X]=-1, regardless of X. Whitened score regularity. From Assumption C.3, the original score satisfies |s(y,η)|≤Cψ1≍1/σ|s(y,η)|≤ C_ _1 1/σ and I(η)≥c−/σ2I(η)≥ c_-/σ^2. Therefore the whitened score satisfies |s~(y,η)|=|s(y,η)|I(η)≤Cψ1c−/σ2=Cψ1σ2c−≤Cs~,| s(y,η)|= |s(y,η)|I(η)≤ C_ _1c_-/σ^2= C_ _1σ^2c_-≤ C_ s, (D.1) where Cs~:=Cψ1σ2/c−C_ s:=C_ _1σ^2/c_- is a bounded constant (since σ is a constant in the BTL model). Similarly, by Assumption C.3(i), |∂ηs~(y,η)|≤|s˙(y,η)|I(η)+|s(y,η)|⋅|I′(η)|I(η)2≤Cs~˙,| _η s(y,η)|≤ | s(y,η)|I(η)+ |s(y,η)|·|I (η)|I(η)^2≤ C_ s, (D.2) and the second derivative |∂η2s~(y,η)|≤Cs~¨|∂^2_η s(y,η)|≤ C_ s, where Cs~˙,Cs~¨C_ s,C_ s are dimension-free constants depending only on (c−,c+,c1,c2,Cψ1,σ)(c_-,c_+,c_1,c_2,C_ _1,σ). In particular, ‖s~(Y,η^)‖ψ1≤Cs~\| s(Y, η)\|_ _1≤ C_ s (the whitened score is bounded, hence sub-exponential with a constant parameter, unlike the original score which has ψ1 _1-norm ≍1/σ 1/σ). Isotropic Gram operator. Under the whitened score and uniform comparison design, by property (i) the effective Gram operator is G0(H)=⋆[⟨H,X⟩X]=1d⋆HG_0(H)=E \! [ H,X \,X ]= 1d \,H (D.3) on the column-sum-zero subspace (Lemma C.31). The restricted operators are therefore A0=PG0P=1d⋆P,A0−1=d⋆P.A_0=P_TG_0P_T= 1d \,P_T, A_0^-1=d \,P_T. Simplified directions. The oracle and estimated EIF directions are Hws⋆=d⋆PΓ,H^ws=d⋆P^Γ,H _ ws=d \,P_T , H_ ws=d \, P_T , (D.4) and the direction error is H^ws−Hws⋆=d⋆(P^−P)Γ. H_ ws-H _ ws=d \,( P_T-P_T) . (D.5) No operator inversion, Neumann series expansion, or Gram mismatch analysis is needed. D.2 Bounding RempH~R_emp H (direction-error empirical process) Define RempH~:=(ℙn−ℙ⋆)[s~(Y,η^)⟨H^ws−Hws⋆,X⟩].R_emp H:=(P_n-P ) [ s(Y, η)\, H_ ws-H _ ws,X ]. Let Zi:=s~(Yi,η^i)⟨H^ws−Hws⋆,Xi⟩Z_i:= s(Y_i, η_i)\, H_ ws-H _ ws,X_i . We bound its variance and sub-exponential norm following the five-step template of Section C.4. D.2.1 Step 1: Variance bound via Frobenius reduction Since |s~(Y,η^)|≤Cs~| s(Y, η)|≤ C_ s (Eq. (D.1)), ⋆[Zi2]≤Cs~2⋆[⟨H^ws−Hws⋆,X⟩2].E [Z_i^2]≤ C_ s^2\,E \! [ H_ ws-H _ ws,X ^2 ]. Applying the Frobenius reduction (Lemma C.31): Var(Zi)≤⋆[Zi2]≤Cs~2⋅‖H^ws−Hws⋆‖F2d⋆.Var(Z_i) [Z_i^2]≤ C_ s^2· H_ ws-H _ ws _F^2d . (D.6) Note that the Fisher weight c+/σ2c_+/σ^2 from the general case is now replaced by the constant Cs~2C_ s^2. D.2.2 Step 2: Extracting ‖Γ‖1 _1 from the Frobenius norm Since H^ws−Hws⋆=d⋆(P^−P)Γ H_ ws-H _ ws=d \,( P_T-P_T) , by Lemma C.33: ‖H^ws−Hws⋆‖F=d⋆‖(P^−P)Γ‖F≤d⋆‖Γ‖1⋅maxω⁡‖(P^−P)Eω‖F. H_ ws-H _ ws _F=d \, ( P_T-P_T) _F≤ d \, _1· _ω ( P_T-P_T)E_ω _F. (D.7) D.2.3 Step 3: Bounding maxω⁡‖(P^−P)Eω‖F _ω ( P_T-P_T)E_ω _F By the subspace estimation guarantee (Eq. (C.18)): maxω⁡‖(P^−P)Eω‖F≤C(μ,r,m)⋅d¯ρd⋆, _ω ( P_T-P_T)E_ω _F≤ C(μ,r,m)· d\,ρ d , (D.8) where ρ=λmin−1d⋆d¯log⁡d¯/nρ= _ ^-1 d d d/n as in (C.35). This is the same bound as in the general case (Eq. (C.18)), with the d¯ d arising from the (I−PU)(I-P_U) mode in the tangent space projector. The key simplification is that no resolvent decomposition (Eq. (C.16)) is needed—the direction error is a single term d⋆(P^−P)Γd ( P_T-P_T) rather than the two-term expansion A−1(P^−P)+A−1(A−A^)A^−1P^A^-1( P_T-P_T)+A^-1(A- A) A^-1 P_T. Substituting (D.8) into (D.7): ‖H^ws−Hws⋆‖F≤d⋆‖Γ‖1⋅C(μ,r,m)⋅d¯ρd⋆=C(μ,r,m)d⋆d¯‖Γ‖1ρ. H_ ws-H _ ws _F≤ d \, _1· C(μ,r,m)· d\,ρ d =C(μ,r,m)\, d d\, _1\,ρ. Plugging into (D.6): Var(Zi)≤C(μ,r,m)d¯‖Γ‖12ρ2.Var(Z_i)≤ C(μ,r,m)\, d\, _1^2\,ρ^2. (D.9) The CA2σ2C_A^2σ^2 prefactor from the general case has been absorbed into C(μ,r,m)C(μ,r,m); the d¯ d factor persists from the uncompressed mode in the tangent projector. D.2.4 Step 4: Sub-exponential (ψ1 _1) bound via ℓ∞ _∞ norm Since |s~|≤Cs~| s|≤ C_ s (bounded, hence sub-exponential with ‖s~‖ψ1≤Cs~\| s\|_ _1≤ C_ s) and |⟨H^ws−Hws⋆,X⟩|≤2‖H^ws−Hws⋆‖∞| H_ ws-H _ ws,X |≤ 2 H_ ws-H _ ws _∞: ‖Zi‖ψ1≤2Cs~‖H^ws−Hws⋆‖∞.\|Z_i\|_ _1≤ 2C_ s\, H_ ws-H _ ws _∞. (D.10) By Lemma C.33 (entrywise version) and the PEωP_TE_ω bound (Lemma C.37): ‖H^ws−Hws⋆‖∞=d⋆‖(P^−P)Γ‖∞≤d⋆‖Γ‖1⋅maxω⁡‖(P^−P)Eω‖∞. H_ ws-H _ ws _∞=d \, ( P_T-P_T) _∞≤ d \, _1· _ω ( P_T-P_T)E_ω _∞. Since ‖(P^−P)Eω‖∞≤‖P^Eω‖∞+‖PEω‖∞≤2C(μ,r,m)d¯/d⋆ ( P_T-P_T)E_ω _∞≤ P_TE_ω _∞+ P_TE_ω _∞≤ 2C(μ,r,m)\, d/d (Lemma C.37): ‖H^ws−Hws⋆‖∞≤d⋆⋅‖Γ‖1⋅2C(μ,r,m)⋅d¯d⋆=C(μ,r,m)d¯‖Γ‖1. H_ ws-H _ ws _∞≤ d · _1· 2C(μ,r,m)· dd =C(μ,r,m)\, d\, _1. (D.11) Therefore: ‖Zi‖ψ1≤C(μ,r,m)d¯‖Γ‖1.\|Z_i\|_ _1≤ C(μ,r,m)\, d\, _1. (D.12) The mechanism is the same as in the general case: d⋆d from the direction cancels with 1/d⋆1/d from incoherence. The simplification is that the whitened score has ψ1 _1-norm O(1)O(1), so the CAσC_Aσ prefactor disappears. D.2.5 Step 5: Bernstein concentration Applying Lemma C.39 with the variance (D.9) and tail bound (D.12): Theorem D.1 (Bound on RempH~R_emp H). Under Assumptions C.3 and the subspace estimation guarantees, with probability at least 1−δ1-δ (conditional on 1D_1): |RempH~|≤C(μ,r,m)∥Γ∥1[d¯ρlog⁡(2/δ)n+d¯log⁡(2/δ)n]. |R_emp H|≤ C(μ,r,m)\, _1 [ d\;ρ\; (2/δ)n\;+\; d\; (2/δ)n ]. (D.13) Proof of Theorem D.1. The variance term gives Var(Zi)log⁡(2/δ)n≤C(μ,r,m)d¯‖Γ‖1ρlog⁡(2/δ)n. Var(Z_i) (2/δ)n≤ C(μ,r,m)\, d\, _1\,ρ\, (2/δ)n. The sub-exponential term gives ‖Zi‖ψ1log⁡(2/δ)n≤C(μ,r,m)d¯‖Γ‖1log⁡(2/δ)n. \|Z_i\|_ _1 (2/δ)n≤ C(μ,r,m)\, d\, _1\, (2/δ)n. Combining yields (D.13). □ D.3 Bounding Rempη~R_emp η (score-perturbation empirical process) Define Rempη~:=(ℙn−ℙ⋆)[(s~(Y,η^)−s~(Y,η⋆))⟨Hws⋆,X⟩].R_emp η:=(P_n-P ) [( s(Y, η)- s(Y,η ))\, H _ ws,X ]. Condition on the first-stage output so that Δ:=T^−T⋆ := T-T and Hws⋆H _ ws are fixed. D.3.1 Taylor expansion of the whitened score For each i, there exists η¯i η_i between η^i η_i and ηi⋆η _i such that s~(Yi,η^i)−s~(Yi,ηi⋆)=∂ηs~(Yi,ηi⋆)⟨Δ,Xi⟩+12∂η2s~(Yi,η¯i)⟨Δ,Xi⟩2. s(Y_i, η_i)- s(Y_i,η _i)= _η s(Y_i,η _i) ,X_i + 12∂^2_η s(Y_i, η_i) ,X_i ^2. Hence Rempη~=Rη~,1emp+Rη~,2empR_emp η=R_ η,1^emp+R_ η,2^emp, where Rη~,1emp:=(ℙn−ℙ⋆)[∂ηs~(Y,η⋆)⟨Δ,X⟩⟨Hws⋆,X⟩],Rη~,2emp:=12(ℙn−ℙ⋆)[∂η2s~(Y,η¯)⟨Δ,X⟩2⟨Hws⋆,X⟩].R_ η,1^emp:=(P_n-P ) [ _η s(Y,η ) ,X H _ ws,X ], R_ η,2^emp:= 12(P_n-P ) [∂^2_η s(Y, η) ,X ^2 H _ ws,X ]. D.3.2 Bounds on ‖Hws⋆‖∞ H _ ws _∞ and ‖Hws⋆‖F/d⋆ H _ ws _F/ d Since Hws⋆=d⋆PΓH _ ws=d \,P_T , by Lemma C.33: ℓ∞ _∞ bound. Using Lemma C.37 (‖PEω‖∞≤C(μ,r,m)d¯/d⋆ P_TE_ω _∞≤ C(μ,r,m)\, d/d ): ‖Hws⋆‖∞≤d⋆‖Γ‖1⋅maxω⁡‖PEω‖∞≤C(μ,r,m)d¯‖Γ‖1. H _ ws _∞≤ d \, _1· _ω P_TE_ω _∞≤ C(μ,r,m)\, d\, _1. (D.14) Compared to (C.29), the factor CAσ2C_Aσ^2 is replaced by 11 (since d⋆d from the direction cancels with 1/d⋆1/d from incoherence directly, without passing through ‖A−1‖∞→∞\|A^-1\|_∞→∞). Frobenius bound. Since ‖PEω‖F2=(P)ωω≤C(μ,r,m)d¯/d⋆ P_TE_ω _F^2=(P_T)_ω≤ C(μ,r,m)\, d/d : ‖Hws⋆‖Fd⋆≤d⋆‖Γ‖1⋅maxω⁡‖PEω‖Fd⋆≤C(μ,r,m)d¯‖Γ‖1. H _ ws _F d ≤ d \, _1· _ω P_TE_ω _F d ≤ C(μ,r,m)\, d\, _1. (D.15) D.3.3 Abstract Bernstein bound The same argument as Proposition C.11 applies with the whitened score derivatives replacing the original ones. Since |∂ηs~|≤Cs~˙| _η s|≤ C_ s and |∂η2s~|≤Cs~¨|∂^2_η s|≤ C_ s (both constants, see (D.2)), the factor 1/σ21/σ^2 in (C.28) is replaced by a constant. Corollary D.2 (Explicit bound for Rempη~R_emp η). Under Assumption C.3 and comparison design, with probability at least 1−δ1-δ, |Rempη~|≤C(μ,r,m)∥Γ∥1(∥Δ∥∞+∥Δ∥∞2)[d¯log⁡(4/δ)n+d¯log⁡(4/δ)n]. |R_emp η|\;≤\;C(μ,r,m)\, _1 ( _∞+ _∞^2 ) [ d (4/δ)n\;+\; d\, (4/δ)n ]. (D.16) Proof of Corollary D.2. Substituting (D.14) and (D.15) into the abstract Bernstein bound (same structure as Proposition C.11 with constant prefactors): the first-order variance term gives Cs~˙⋅‖Δ‖∞⋅‖Hws⋆‖Fd⋆≤C(μ,r,m)‖Γ‖1‖Δ‖∞d¯;C_ s· _∞· H _ ws _F d ≤ C(μ,r,m)\, _1\, _∞ d; the first-order tail term gives Cs~˙⋅‖Δ‖∞⋅‖Hws⋆‖∞≤C(μ,r,m)‖Γ‖1‖Δ‖∞d¯;C_ s· _∞· H _ ws _∞≤ C(μ,r,m)\, _1\, _∞\, d; the second-order terms are the same with an extra factor of ‖Δ‖∞ _∞. Factoring ‖Δ‖∞+‖Δ‖∞2 _∞+ _∞^2 yields (D.16). □ D.4 Bounding RprojR_proj (projection leakage) The projection leakage term is Rproj:=⟨(I−P)Γ,Δ⟩.R_proj:= (I-P_T) , . This term depends only on the functional gradient Γ and the estimation error Δ=T^−T⋆ = T-T ; it is independent of the score function or the direction H. Therefore, the analysis of Section C.6 applies verbatim. By Proposition C.13 and the initialization guarantees: |Rproj|≤C(μ,r,m)∥Γ∥1ρ2. |R_proj|\;≤\;C(μ,r,m)\; _1\,ρ^2. (D.17) This is identical to (C.38). D.5 H-direction bias The H-direction bias term is ℙ⋆[s~(Y,η^)⟨H^ws−Hws⋆,X⟩].P [ s(Y, η) H_ ws-H _ ws,X ]. D.5.1 Taylor expansion and leading term Expanding s~(Y,η^) s(Y, η) around η⋆η (as in Section C.7): ℙ⋆[s~(Y,η^)⟨H^ws−Hws⋆,X⟩] [ s(Y, η) H_ ws-H _ ws,X ] =ℙ⋆[s~(Y,η⋆)⟨H^ws−Hws⋆,X⟩]⏟= 0 (centering) = P [ s(Y,η ) H_ ws-H _ ws,X ]_=\,0 (centering) +ℙ⋆[∂ηs~(Y,η⋆)⟨Δ,X⟩⟨H^ws−Hws⋆,X⟩]⏟first-order in Δ + P [ _η s(Y,η ) ,X H_ ws-H _ ws,X ]_first-order in +12ℙ⋆[∂η2s~(Y,η¯)⟨Δ,X⟩2⟨H^ws−Hws⋆,X⟩]⏟second-order in Δ. + 12P [∂^2_η s(Y, η) ,X ^2 H_ ws-H _ ws,X ]_second-order in . (D.18) D.5.2 First-order piece By property (i) of the whitened score, ⋆[∂ηs~(Y,η⋆)∣X]=−1E [ _η s(Y,η ) X]=-1. Therefore the first-order piece equals −⟨Δ,(H^ws−Hws⋆)/d⋆⟩- ,( H_ ws-H _ ws)/d computed via the isotropic Gram: ℙ⋆[∂ηs~⟨Δ,X⟩⟨H^ws−Hws⋆,X⟩]=−⟨Δ,G0(H^ws−Hws⋆)⟩=−1d⋆⟨Δ,H^ws−Hws⋆⟩.P [ _η s\, ,X H_ ws-H _ ws,X ]=- ,G_0( H_ ws-H _ ws) =- 1d , H_ ws-H _ ws . Since H^ws−Hws⋆=d⋆(P^−P)Γ H_ ws-H _ ws=d ( P_T-P_T) : −1d⋆⟨Δ,d⋆(P^−P)Γ⟩=−⟨Δ,(P^−P)Γ⟩.- 1d ,d ( P_T-P_T) =- ,( P_T-P_T) . Decomposing Δ=PΔ+(I−P)Δ =P_T +(I-P_T) : • By Cauchy–Schwarz, |⟨PΔ,(P^−P)Γ⟩|≤‖PΔ‖F⋅‖(P^−P)Γ‖F.| P_T ,( P_T-P_T) |≤ P_T _F· ( P_T-P_T) _F. Using ‖(P^−P)Γ‖F≤‖Γ‖1⋅C(μ,r,m)d¯ρ/d⋆ ( P_T-P_T) _F≤ _1· C(μ,r,m) d\,ρ/ d (from (D.8)) and ‖PΔ‖F/d⋆≤‖Δ‖∞ P_T _F/ d ≤ _∞, this contributes O(d¯‖Δ‖∞⋅ρ⋅‖Γ‖1)O( d\, _∞·ρ· _1), which is an O(‖Δ‖∞⋅ρ)O( _∞·ρ) cross term retained in the combined bound. • By Hölder’s inequality, |⟨(I−P)Δ,(P^−P)Γ⟩|≤‖(I−P)Δ‖∞⋅‖(P^−P)Γ‖1.| (I-P_T) ,( P_T-P_T) |≤ (I-P_T) _∞· ( P_T-P_T) _1. By Lemma C.15, ‖(P^−P)Γ‖1≤‖Γ‖1⋅C(μ,r,m) ( P_T-P_T) _1≤ _1· C(μ,r,m), and by Proposition C.13, ‖(I−P)Δ‖∞≤C(μ,r,m)ρ2 (I-P_T) _∞≤ C(μ,r,m)ρ^2. This gives O(ρ2⋅‖Γ‖1)O(ρ^2· _1). D.5.3 Second-order piece Since |∂η2s~|≤Cs~¨|∂^2_η s|≤ C_ s (a constant) and |⟨Δ,X⟩|≤2‖Δ‖∞| ,X |≤ 2 _∞: |12ℙ⋆[∂η2s~⟨Δ,X⟩2⟨H^ws−Hws⋆,X⟩]|≤Cs~¨‖Δ‖∞2⋆[|⟨H^ws−Hws⋆,X⟩|]. | 12P [∂^2_η s\, ,X ^2 H_ ws-H _ ws,X ] |≤ C_ s\, _∞^2\,E [| H_ ws-H _ ws,X |]. By the ℓ1 _1-average bound (Lemma C.14): ⋆[|⟨H^ws−Hws⋆,X⟩|]≤‖H^ws−Hws⋆‖1d⋆.E [| H_ ws-H _ ws,X |]≤ H_ ws-H _ ws _1d . Since H^ws−Hws⋆=d⋆(P^−P)Γ H_ ws-H _ ws=d ( P_T-P_T) and ‖(P^−P)Γ‖1≤‖Γ‖1⋅C(μ,r,m) ( P_T-P_T) _1≤ _1· C(μ,r,m) (by Lemma C.15): ‖H^ws−Hws⋆‖1d⋆=‖(P^−P)Γ‖1≤C(μ,r,m)‖Γ‖1. H_ ws-H _ ws _1d = ( P_T-P_T) _1≤ C(μ,r,m)\, _1. Therefore the second-order piece satisfies: |second-order H-bias|≤C(μ,r,m)∥Δ∥∞2∥Γ∥1. |second-order H-bias |≤ C(μ,r,m)\, _∞^2\, _1. (D.19) The key cancellation is identical to the general case (Eq. (C.46)): d⋆d from the direction cancels with 1/d⋆1/d from the ℓ1 _1-average bound. The prefactor 1/σ21/σ^2 from |s¨|| s| is absent here because |∂η2s~||∂^2_η s| is bounded by a constant. D.6 First-order cancellation The combined first-order cancellation term from the bias decomposition is (same as Eq. (C.47)): ⟨PΓ,Δ⟩+ℙ⋆[s~(Y,η^)⟨Hws⋆,X⟩]−ℙ⋆[s~(Y,η⋆)⟨Hws⋆,X⟩]. P_T , +P [ s(Y, η) H _ ws,X ]-P [ s(Y,η ) H _ ws,X ]. (D.20) D.6.1 Taylor expansion of the population score difference The second term of (D.20) minus the third (which vanishes by centering) gives, by Taylor expansion around η⋆η : ℙ⋆[s~(Y,η^)⟨Hws⋆,X⟩]=−⟨Δ,G0(Hws⋆)⟩+R~2(Δ;Hws⋆),P [ s(Y, η) H _ ws,X ]=- ,G_0(H _ ws) + R_2( ;H _ ws), (D.21) where we used ⋆[∂ηs~(Y,η⋆)∣X]=−1E [ _η s(Y,η ) X]=-1 (property (i)) to get G0(H)=(1/d⋆)HG_0(H)=(1/d )H, and R~2(Δ;Hws⋆):=12ℙ⋆[∂η2s~(Y,η¯)⟨Δ,X⟩2⟨Hws⋆,X⟩]. R_2( ;H _ ws):= 12P [∂^2_η s(Y, η) ,X ^2 H _ ws,X ]. D.6.2 The cancellation mechanism — complete cancellation Plugging (D.21) into (D.20): ⟨PΓ,Δ⟩−⟨Δ,G0(Hws⋆)⟩+R~2(Δ;Hws⋆). P_T , - ,G_0(H _ ws) + R_2( ;H _ ws). Now G0(Hws⋆)=(1/d⋆)⋅d⋆PΓ=PΓG_0(H _ ws)=(1/d )· d P_T =P_T . Therefore: ⟨PΓ,Δ⟩−⟨Δ,PΓ⟩=0. P_T , - ,P_T =0. (D.22) This is the key structural advantage of score whitening: in the general case (Section C.8), the first-order term ⟨PΓ,Δ⟩−⟨Δ,G(H⋆)⟩ P_T , - ,G(H ) leaves a surviving normal-component contribution −⟨(I−P)Δ,G(H⋆)⟩- (I-P_T) ,G(H ) that requires O(CA⋅‖(I−P)Δ‖∞)O(C_A· (I-P_T) _∞) to control (Proposition C.18). Under score whitening, G0(Hws⋆)=PΓ∈G_0(H _ ws)=P_T , so ⟨(I−P)Δ,G0(Hws⋆)⟩=⟨(I−P)Δ,PΓ⟩=0 (I-P_T) ,G_0(H _ ws) = (I-P_T) ,P_T =0 by orthogonality of (I−P)Δ∈⟂(I-P_T) and PΓ∈P_T . Only the second-order remainder R~2 R_2 survives. D.7 Second-order score remainder The second-order remainder from (D.21) is R~2(Δ;Hws⋆)=12ℙ⋆[∂η2s~(Y,η¯)⟨Δ,X⟩2⟨Hws⋆,X⟩]. R_2( ;H _ ws)= 12P [∂^2_η s(Y, η) ,X ^2 H _ ws,X ]. Proposition D.3 (Second-order remainder bound). Under the comparison design, with |∂η2s~|≤Cs~¨|∂^2_η s|≤ C_ s, |R~2(Δ;Hws⋆)|≤Cs~¨‖Δ‖∞2‖Hws⋆‖1d⋆.| R_2( ;H _ ws)|≤ C_ s\, _∞^2\, H _ ws _1d . (D.23) Proof of Proposition D.3. Since |⟨Δ,X⟩|≤2‖Δ‖∞| ,X |≤ 2 _∞ (comparison atom) and |∂η2s~|≤Cs~¨|∂^2_η s|≤ C_ s: |R~2|≤Cs~¨‖Δ‖∞2⋆[|⟨Hws⋆,X⟩|].| R_2|≤ C_ s\, _∞^2\,E [| H _ ws,X |]. By the ℓ1 _1-average bound (Lemma C.14), ⋆[|⟨Hws⋆,X⟩|]≤‖Hws⋆‖1/d⋆E [| H _ ws,X |]≤ H _ ws _1/d . □ Expanding ‖Hws⋆‖1/d⋆ H _ ws _1/d . Since Hws⋆=d⋆PΓH _ ws=d P_T : ‖Hws⋆‖1d⋆=‖PΓ‖1≤‖Γ‖1⋅maxω⁡‖PEω‖1≤C(μ,r,m)‖Γ‖1, H _ ws _1d = P_T _1≤ _1· _ω P_TE_ω _1≤ C(μ,r,m)\, _1, where the last step uses Lemma C.15 (‖PEω‖1≤CP=C(μ,r,m) P_TE_ω _1≤ C_P=C(μ,r,m), dimension-free). Therefore: |R~2(Δ;Hws⋆)|≤C(μ,r,m)∥Δ∥∞2∥Γ∥1. | R_2( ;H _ ws)|≤ C(μ,r,m)\, _∞^2\, _1. (D.24) This matches (C.53) in form but without CAC_A: the d⋆d from the direction cancels with 1/d⋆1/d from the ℓ1 _1-average bound, and the whitened score derivative is O(1)O(1) rather than O(1/σ2)O(1/σ^2). D.8 Combined bound We now collect all remainder terms. D.8.1 Recap of individual bounds The total remainder RnR_n decomposes as: Rn=RempH~⏟Sec. D.2+Rempη~⏟Sec. D.3+Rproj⏟Sec. D.4+H-bias⏟Sec. D.5+R~2(Δ;Hws⋆)⏟Sec. D.7.R_n= R_emp H_Sec.~ app:ws-RempH+ R_emp η_Sec.~ app:ws-Remp-eta+ R_proj_Sec.~ app:ws-Rproj+ H-bias_Sec.~ app:ws-H-bias+ R_2( ;H _ ws)_Sec.~ app:ws-2nd-order. (D.25) The individual bounds are: (i) Direction-error empirical process (Theorem D.1): |RempH~|≤C(μ,r,m)‖Γ‖1[d¯ρlog⁡(2/δ)n+d¯log⁡(2/δ)n].|R_emp H|≤ C(μ,r,m)\, _1 [ d\,ρ (2/δ)n+ d\, (2/δ)n ]. (i) Score-perturbation empirical process (Corollary D.2): |Rempη~|≤C(μ,r,m)‖Γ‖1(‖Δ‖∞+‖Δ‖∞2)[d¯n+d¯n]|R_emp η|≤ C(μ,r,m)\, _1 ( _∞+ _∞^2 ) [ dn+ dn ] (up to logarithmic factors). (i) Projection leakage (Eq. (D.17)): |Rproj|≤C(μ,r,m)‖Γ‖1ρ2.|R_proj|≤ C(μ,r,m)\, _1\,ρ^2. (iv) H-direction bias (Section D.5): first order≤C(μ,r,m)‖Γ‖1(d¯‖Δ‖∞⋅ρ+ρ2),second order≤C(μ,r,m)‖Δ‖∞2‖Γ‖1.first order≤ C(μ,r,m)\, _1 ( d\, _∞·ρ+ρ^2 ), order≤ C(μ,r,m)\, _∞^2\, _1. (v) First-order cancellation and second-order remainder (Sections D.6–D.7): first order cancels completely (Eq. (D.22)); the second-order remainder satisfies |R~2|≤C(μ,r,m)‖Δ‖∞2‖Γ‖1.| R_2|≤ C(μ,r,m)\, _∞^2\, _1. D.8.2 Combined bound Theorem D.4 (Combined remainder bound for the score-whitened estimator). Under Assumption C.3 and the subspace estimation guarantees (P1)–(P5), with probability at least 1−δ1-δ, |Rn|≤C(μ,r,m)∥Γ∥1[ |R_n|\;≤\;C(μ,r,m)\, _1 [ d¯ρlog⁡(2/δ)n+d¯log⁡(2/δ)n+ d\;ρ (2/δ)n\;+\; d\, (2/δ)n\;+\; (D.26) ∥Δ∥∞d¯n+∥Δ∥∞ρ+ρ2+∥Δ∥∞2], _∞ dn\;+\; _∞\,ρ\;+\;ρ^2+ _∞^2 ], where ρ=σλmin−1d⋆d¯log⁡d¯/nρ=σ _ ^-1 d d d/n. Under the SNR condition, ρ≍‖Δ‖∞≍d¯/nρ _∞ d/n (up to logarithmic factors), so all terms are the same order; none is negligible. Remark D.5 (Simplified bound for pairwise comparisons). In the pairwise-comparison setting (σ=O(1)σ=O(1), λmin≍d⋆ _ d ), substituting ρ≍d¯logc⁡d¯/nρ d ^c d/n, ‖Δ‖∞≲d¯logc⁡d¯/n _∞ d ^c d/n, and δ=d¯−cδ= d^-c into (D.26) gives |Rn|≤C(μ,r,m)‖Γ‖1d¯logc⁡d¯n |R_n|\;≤\;C(μ,r,m)\, _1\, d ^c dn (D.27) with probability ≥1−d¯−c≥ 1- d^-c. This is the bound stated in Theorem 5.1. Crucially, no CAC_A appears: the score-whitening construction eliminates the operator inverse entirely. Proof of Theorem D.4. Sum the bounds (i)–(v): • Term (i) contributes d¯ρlog/n d\,ρ /n (variance) and d¯log/n d /n (sub-exponential tail). • Term (i) contributes ‖Δ‖∞d¯/n _∞ d/n (first-order piece) and ‖Δ‖∞2d¯/n _∞^2 d/n (second-order, lower order). • Term (i) contributes ρ2ρ^2. • Term (iv) contributes d¯‖Δ‖∞⋅ρ d\, _∞·ρ (first order) and ‖Δ‖∞2 _∞^2 (second order). • Term (v): first order cancels completely; second-order remainder contributes ‖Δ‖∞2 _∞^2. Combining yields (D.26). □ Appendix E Proofs for IPW and Nonlinear Extensions This section provides the detailed proofs for the IPW and nonlinear extensions in Section 5: IPW with known weights (§E.1), IPW with estimated weights (§E.2), and nonlinear functionals (§E.3). Throughout, we work under the comparison design and Assumption C.3. E.1 IPW with known weights We verify that the five-term decomposition of Appendix D carries through when the whitened score s~ s is replaced by the IPW-whitened score s~w(y,η;x):=w(x)s~(y,η) s^w(y,η;x):=w(x) s(y,η), where w(x)=q(x)/p(x)w(x)=q(x)/p(x) and q is the uniform reference distribution. E.1.1 Setup The key structural fact, established in the main text (5.12), is that the effective Gram operator under the IPW score satisfies Gq(H)=X∼p[w(X)⟨H,X⟩X]=X∼q[⟨H,X⟩X]=1d⋆HG_q(H)=E_X p\! [w(X) H,X X ]=E_X q\! [ H,X X ]= 1d H on the column-sum-zero subspace. This is identical to G0=(1/d⋆)IG_0=(1/d )I from Appendix D. The oracle and estimated directions are therefore the same as (5.13): Hq,0=d⋆PΓ,H^q,0=d⋆P^Γ,H_q,0=d \,P_T , H_q,0=d \, P_T , which coincide with Hws⋆H _ ws and H^ws H_ ws. E.1.2 Remainder decomposition The total remainder RnipwR_n ipw for the IPW estimator (5.14) decomposes exactly as (D.25): Rnipw=RempH~,w+Rempη~,w+Rproj+H-biasw+R~2w(Δ;Hws⋆),R_n ipw=R_emp H,w+R_emp η,w+R_proj+H-bias^w+ R_2^w( ;H _ ws), (E.1) where each “w”-superscripted term is the same as the corresponding term in Appendix D with s~ s replaced by w(X)s~w(X) s. E.1.3 Effect of the weight factor Each random variable in the decomposition acquires at most an additional multiplicative factor of |w(X)||w(X)|. Under the overlap condition (5.15), ∥w∥∞≤Cp/cp=:Cw\|w\|_∞≤ C_p/c_p=:C_w, which is a dimension-free constant. Specifically: (i) Direction-error empirical process RempH~,wR_emp H,w: The summand Ziw=w(Xi)s~(Yi,ηi⋆)⟨H^ws−Hws⋆,Xi⟩Z_i^w=w(X_i) s(Y_i,η _i) H_ ws-H _ ws,X_i satisfies ‖Ziw‖ψ1≤Cw‖Zi‖ψ1,Var(Ziw)≤Cw2Var(Zi).\|Z_i^w\|_ _1≤ C_w\,\|Z_i\|_ _1, (Z_i^w)≤ C_w^2\,Var(Z_i). Therefore Theorem D.1 applies with C(μ,r,m)C(μ,r,m) replaced by C(μ,r,m,Cw)C(μ,r,m,C_w). (i) Score-perturbation Rempη~,wR_emp η,w: The Taylor expansion of w(X)s~(Y,η^)w(X) s(Y, η) around η⋆η produces the same structure as Section D.3 with each derivative ∂ηks~ _η^k s multiplied by w(X)w(X). Since |w(X)∂ηks~|≤Cw|∂ηks~||w(X) _η^k s|≤ C_w| _η^k s|, Corollary D.2 holds with a constant-factor change. (i) Projection leakage RprojR_proj: This term is independent of the score (Eq. (D.17)); it remains unchanged. (iv) H-direction bias: The population expectation is now p[w(X)s~(Y,η^)⟨H^ws−Hws⋆,X⟩]=q[s~(Y,η^)⟨H^ws−Hws⋆,X⟩]E_p[w(X) s(Y, η) H_ ws-H _ ws,X ]=E_q[ s(Y, η) H_ ws-H _ ws,X ], which is the uniform-design expectation. All bounds from Section D.5 apply verbatim (the change of measure from p to q via w is exact, not approximate). (v) First-order cancellation and second-order remainder: The first-order cancellation uses p[w(X)∂ηs~(Y,η⋆)∣X]=−w(X)E_p[w(X) _η s(Y,η ) X]=-w(X) and the Gram identity Gq=G0=(1/d⋆)IG_q=G_0=(1/d )I, giving the same complete cancellation (D.22). The second-order remainder acquires at most a factor CwC_w: |R~2w|≤CwC(μ,r,m)‖Δ‖∞2‖Γ‖1| R_2^w|≤ C_w\,C(μ,r,m)\, _∞^2\, _1. Corollary E.1 (Combined IPW remainder bound). Under Assumption C.3, the overlap condition (5.15) with known weights, and the subspace estimation guarantees (P1)–(P5), with probability at least 1−δ1-δ: |Rnipw|≤C(μ,r,m,Cw)∥Γ∥1[ |R_n ipw|\;≤\;C(μ,r,m,C_w)\, _1 [ d¯ρlog⁡(2/δ)n+d¯log⁡(2/δ)n+ d\;ρ (2/δ)n\;+\; d\, (2/δ)n\;+\; (E.2) ∥Δ∥∞d¯n+∥Δ∥∞ρ+ρ2+∥Δ∥∞2]. _∞ dn\;+\; _∞\,ρ\;+\;ρ^2+ _∞^2 ]. This is identical to Theorem D.4 with C(μ,r,m)C(μ,r,m) replaced by C(μ,r,m,Cw)C(μ,r,m,C_w), where Cw=Cp/cpC_w=C_p/c_p depends only on the overlap constants. Proof. Sum the five individual bounds (i)–(v) above. Each acquires at most Cw2C_w^2 (for variance) or CwC_w (for tail/bias) compared to the corresponding bound in Theorem D.4. All structural cancellations (isotropic Gram, complete first-order cancellation) are preserved. □ E.2 IPW with estimated weights Suppose the estimated sampling distribution p p satisfies the entrywise relative error guarantee δw:=‖w^−w‖∞≤Cwϵp,ϵp≪ 1, _w\;:=\;\| w-w\|_∞\;≤\;C_w\, _p, _p\; \;1, (E.3) where w^(x):=q(x)/p^(x) w(x):=q(x)/ p(x) and Cw:=Cp/cpC_w:=C_p/c_p is the overlap ratio. E.2.1 Decomposition The feasible IPW remainder decomposes as Rnipw,w^=Rnipw⏟known-weight+RnΔw⏟weight-estimation error,R_n ipw, w\;=\; R_n ipw_known-weight\;+\; R_n w_weight-estimation error, (E.4) where RnipwR_n ipw is bounded in Corollary E.1. The weight-estimation remainder further splits as RnΔw=BΔw⏟population bias+RempΔw⏟empirical process.R_n w\;=\; B w_population bias\;+\; R_emp w_empirical process. (E.5) E.2.2 Population bias The leading-order weight error vanishes by score centering: ℙ⋆[(w^(X)−w(X))s~(Y,η⋆)⟨Hq,0⋆,X⟩]= 0.P [( w(X)-w(X))\, s(Y,η )\, H _q,0,X ]\;=\;0. (E.6) What remains is driven by the score perturbation s~(Y,η^)−s~(Y,η⋆) s(Y, η)- s(Y,η ) and the direction error H^q,0−Hq,0⋆ H_q,0-H _q,0, both of which are second-order. First component. By Taylor expansion and the complete first-order cancellation (D.22): |ℙ⋆[s~(Y,η^)⟨Hq,0⋆,X⟩]|≤C(μ,r,m)‖Δ‖∞2‖Γ‖1. |P [ s(Y, η)\, H _q,0,X ] |\;≤\;C(μ,r,m)\, _∞^2\, _1. Second component. By the ℓ1 _1-average bound (Lemma C.14) and the direction-error estimate: |ℙ⋆[s~(Y,η^)⟨H^q,0−Hq,0⋆,X⟩]|≤Cs~‖H^q,0−Hq,0⋆‖1d⋆≤C(μ,r,m)‖Γ‖1(‖Δ‖∞ρ+ρ2). |P [ s(Y, η)\, H_q,0-H _q,0,X ] |\;≤\;C_ s\, H_q,0-H _q,0 _1d \;≤\;C(μ,r,m)\, _1\,( _∞\,ρ+ρ^2). Combined bias bound. |BΔw|≤C(μ,r,m,Cw)δw‖Γ‖1[‖Δ‖∞2+‖Δ‖∞ρ+ρ2].|B w|\;≤\;C(μ,r,m,C_w)\, _w\, _1\, [ _∞^2+ _∞\,ρ+ρ^2 ]. (E.7) E.2.3 Empirical process The empirical process term is RempΔw=(ℙn−ℙ⋆)[(w^(X)−w(X))s~(Y,η^)⟨H^q,0,X⟩].R_emp w\;=\;(P_n-P ) [( w(X)-w(X))\, s(Y, η)\, H_q,0,X ]. Factoring out δw _w and using the Frobenius reduction: Variance bound. Var((w^−w)s~⟨H^q,0,X⟩)≤C(μ,r,m,Cw)δw2d¯‖Γ‖12.Var (( w-w) s\, H_q,0,X )\;≤\;C(μ,r,m,C_w)\, _w^2\, d\, _1^2. (E.8) Sub-exponential norm. ‖(w^−w)s~⟨H^q,0,X⟩‖ψ1≤C(μ,r,m,Cw)δwd¯‖Γ‖1. \|( w-w) s\, H_q,0,X \|_ _1\;≤\;C(μ,r,m,C_w)\, _w\, d\, _1. (E.9) Bernstein bound. By Lemma C.39, with probability at least 1−δ1-δ: |RempΔw|≤C(μ,r,m,Cw)δw‖Γ‖1[d¯log⁡(2/δ)n+d¯log⁡(2/δ)n].|R_emp w|\;≤\;C(μ,r,m,C_w)\, _w\, _1 [ d (2/δ)n\;+\; d\, (2/δ)n ]. (E.10) Since δw≤Cwϵp=O(dpar/n) _w≤ C_w _p=O( d_par/n) with dpar≪d⋆d_par d , this is lower order than the known-weight terms. E.2.4 Combined bound Theorem E.2 (Feasible IPW remainder bound). Under Assumption C.3, the overlap condition, and the weight estimation guarantee (E.3) with ϵp≤1/2 _p≤ 1/2, with probability at least 1−δ1-δ: |Rnipw,w^|≤C(μ,r,m,Cw)‖Γ‖1 |R_n ipw, w|\;≤\;C(μ,r,m,C_w)\, _1 [d¯ρlog⁡(2/δ)n+d¯log⁡(2/δ)n+ [ d\;ρ (2/δ)n\;+\; d\, (2/δ)n\;+\; (E.11) ∥Δ∥∞d¯n+∥Δ∥∞ρ+ρ2+∥Δ∥∞2+ϵp∥Δ∥∞]. _∞ dn\;+\; _∞\,ρ\;+\;ρ^2+ _∞^2+ _p _∞ ]. The weight-estimation term ϵp‖Δ‖∞ _p _∞ is comparable to ‖Δ‖∞2 _∞^2 when dpar≪d⋆d_par d . Proof. By (E.4), Rnipw,w^=Rnipw+RnΔwR_n ipw, w=R_n ipw+R_n w. The first term is bounded by Corollary E.1. For the second: the population bias (E.7) contributes O(ϵp)O( _p) times the existing second-order terms; the empirical process (E.10) contributes δw _w times the leading empirical terms. Both carry an extra factor ϵp=O(dpar/n) _p=O( d_par/n) and are lower order. □ E.3 Nonlinear functionals We prove Theorem 5.5 by reducing the nonlinear problem to the linear score-whitened theorem (Theorem D.4) plus two explicit second-order corrections. E.3.1 Setup and exact decomposition Let Γψ:=∇ψ(T⋆) _ψ:=∇ψ(T ), Γ^ψ:=∇ψ(T^) _ψ:=∇ψ( T), and Δ:=T^−T⋆ := T-T . Decompose the plug-in direction as H^ψ=d⋆P^Γψ⏟=⁣:H^ψ(a)+d⋆P^(Γ^ψ−Γψ)⏟=⁣:H^ψ(b), H_ψ= d P_T _ψ_=: H_ψ^(a)+ d P_T( _ψ- _ψ)_=: H_ψ^(b), (E.12) and define the linearized estimator with the true gradient: ψ^lin,ψ:=⟨Γψ,T^⟩+ℙn[s~(Y,η^)⟨H^ψ(a),X⟩]. ψ_lin,ψ:= _ψ, T +P_n [ s(Y, η) H_ψ^(a),X ]. Adding and subtracting ψ^lin,ψ ψ_lin,ψ gives the exact identity ψ^nl−ψ(T⋆)=(ψ^lin,ψ−⟨Γψ,T⋆⟩)⏟linearized estimator+Rplug,ψ⏟plug-in Taylor remainder+R∇,ψ⏟gradient perturbation, ψ_ nl-ψ(T )\;=\; ( ψ_lin,ψ- _ψ,T )_linearized estimator\;+\; R_plug,ψ_plug-in Taylor remainder\;+\; R_∇,ψ_gradient perturbation, (E.13) where Rplug,ψ:=ψ(T^)−ψ(T⋆)−⟨Γψ,Δ⟩,R∇,ψ:=ℙn[s~(Y,η^)⟨H^ψ(b),X⟩].R_plug,ψ:=ψ( T)-ψ(T )- _ψ, , R_∇,ψ:=P_n [ s(Y, η) H_ψ^(b),X ]. E.3.2 Term 1: Linearized estimator (leading term) The first term in (E.13) is exactly the linear score-whitened estimator with the deterministic gradient Γψ _ψ replacing Γ . By Theorem D.4: ψ^lin,ψ−⟨Γψ,T⋆⟩=(ℙn−ℙ⋆)ϕ~ψ⋆+Rnlin(Γψ), ψ_lin,ψ- _ψ,T \;=\;(P_n-P ) φ_ψ +R_n^lin( _ψ), (E.14) where ϕ~ψ⋆(X,Y):=s~(Y,η⋆)⟨d⋆PΓψ,X⟩ φ_ψ (X,Y):= s(Y,η ) d P_T _ψ,X and |Rnlin(Γψ)|≤C(μ,r,m)‖Γψ‖1Bn,|R_n^lin( _ψ)|\;≤\;C(μ,r,m)\, _ψ _1\,B_n, (E.15) with Bn:=d¯ρlog⁡(2/δ)/n+d¯log⁡(2/δ)/n+‖Δ‖∞d¯/n+‖Δ‖∞ρ+ρ2+‖Δ‖∞2B_n:= d\,ρ\, (2/δ)/n+ d\, (2/δ)/n+ _∞ d/n+ _∞\,ρ+ρ^2+ _∞^2. Since Γψ _ψ is supported on SψS_ψ with |Sψ|=sψ=O(1)|S_ψ|=s_ψ=O(1), we have ‖Γψ‖1≤sψ‖Γψ‖F=O(1) _ψ _1≤ s_ψ\,\| _ψ\|_F=O(1), so the linearized remainder is of the same order as in the linear problem. E.3.3 Term 2: Plug-in Taylor remainder Under Assumption 5.4, ψ(T)=g(TSψ)ψ(T)=g(T_S_ψ) for a twice differentiable g:ℝsψ→ℝg:R^s_ψ . Write z⋆:=TSψ⋆z :=T _S_ψ and z^:=T^Sψ=z⋆+ΔSψ z:= T_S_ψ=z + _S_ψ. By the second-order Taylor formula in ℝsψR^s_ψ, there exists z~ z on the segment joining z⋆z and z z such that Rplug,ψ=g(z^)−g(z⋆)−∇g(z⋆)⊤(z^−z⋆)=12ΔSψ⊤∇2g(z~)ΔSψ.R_plug,ψ\;=\;g( z)-g(z )-∇ g(z ) ( z-z )\;=\; 12\, _S_ψ \,∇^2g( z)\, _S_ψ. Using ‖∇2g(z~)‖op≤Lg\|∇^2g( z)\|_ op≤ L_g and ‖ΔSψ‖22≤sψ‖Δ‖∞2\| _S_ψ\|_2^2≤ s_ψ _∞^2: |Rplug,ψ|≤12Lgsψ∥Δ∥∞2≲∥Δ∥∞2. |R_plug,ψ|\;≤\; 12L_g\,s_ψ\, _∞^2\; \; _∞^2. (E.16) This is genuinely second-order because ψ depends on only sψ=O(1)s_ψ=O(1) entries: the finite-support condition ensures that the Taylor remainder involves only a bounded-dimensional Hessian applied to a vector of length ‖Δ‖∞ _∞. E.3.4 Term 3: Gradient perturbation The gradient perturbation direction is H^ψ(b)=d⋆P^(Γ^ψ−Γψ) H_ψ^(b)=d P_T( _ψ- _ψ). Since Γ^ψ−Γψ _ψ- _ψ is supported on SψS_ψ and the mean-value theorem gives ‖Γ^ψ−Γψ‖∞≤Lg‖Δ‖∞ _ψ- _ψ _∞≤ L_g _∞: ‖Γ^ψ−Γψ‖1≤sψLg‖Δ‖∞≲‖Δ‖∞. _ψ- _ψ _1\;≤\;s_ψ\,L_g\, _∞\; \; _∞. (E.17) The term R∇,ψR_∇,ψ has the same structure as the linear score-whitened one-step correction with Γ^ψ−Γψ _ψ- _ψ replacing Γ . Applying the linear theorem: |R∇,ψ|≤C(μ,r,m,Lg,sψ)‖Δ‖∞Bn.|R_∇,ψ|\;≤\;C(μ,r,m,L_g,s_ψ)\, _∞\,B_n. (E.18) This is O(‖Δ‖∞)O( _∞) times the linearized remainder (E.15) and hence strictly lower order. E.3.5 Combined bound Theorem E.3 (Remainder bound for nonlinear one-step estimator). Under Assumption C.3, Assumption 5.4, and the subspace estimation guarantees (P1)–(P5), with probability at least 1−δ1-δ: |Rnnl|≤C(μ,r,m,Lg,sψ)[∥Γψ∥1Bn+∥Δ∥∞Bn+∥Δ∥∞2], |R_n nl|\;≤\;C(μ,r,m,L_g,s_ψ) [ _ψ _1\,B_n\;+\; _∞\,B_n\;+\; _∞^2 ], (E.19) where BnB_n is as in (E.15). Proof. Sum the three terms from (E.13): the linearized remainder (E.15), the plug-in Taylor remainder (E.16), and the gradient perturbation (E.18). Since ‖Γψ‖1=O(1) _ψ _1=O(1), the linearized term dominates. The gradient-perturbation term carries an extra factor ‖Δ‖∞=o(1) _∞=o(1) and is lower order. The plug-in Taylor remainder is O(‖Δ‖∞2)=O(d¯/n)O( _∞^2)=O( d/n), which is also lower order. □ CLT. Under the pairwise specialization ‖Δ‖∞≲d¯logc⁡d¯/n _∞ d ^c d/n, all three remainder terms are O(d¯logc⁡d¯/n)O( d ^c d/n), giving nRnnl→0 nR_n nl→ 0 under n≫d¯logc⁡d¯n d ^c d. Hence n(ψ^nl−ψ(T⋆))→(0,Vws(ψ)),Vws(ψ)=⋆[⟨d⋆PΓψ,X⟩2I(η⋆)]. n ( ψ_ nl-ψ(T ) )\; d\;N(0,\;V_ ws(ψ)), V_ ws(ψ)=E \! [ d P_T _ψ,X ^2I(η ) ]. Appendix F Berry–Esseen Bound for the Pairwise-Comparison Estimators This section establishes a quantitative Berry–Esseen bound for the one-step estimators. The CLT convergence itself follows from a standard conditional Lyapunov CLT and Slutsky’s theorem once the remainder conditions of Theorems 4.4 and 5.2 are verified (as sketched in the main text). Here we go further and derive an explicit rate of convergence for the Kolmogorov distance. The argument proceeds in two stages: first, we bound the Berry–Esseen error of the leading i.i.d. term (Subsection F.1); then we quantify the additional error contributed by the one-step remainder RnR_n (Subsection F.4). We use the following regularity conditions throughout, all of which hold under the bounded-signal condition ‖T⋆‖∞≤B\|T \|_∞≤ B of the main text: (R1) Fisher information bounds: there exist 0<cI≤CI<∞0<c_I≤ C_I<∞ such that cI≤I(η⋆)≤CIc_I≤ I(η )≤ C_I uniformly over all admissible X; (R2) Third-moment bounds: there exist M3,ws,M3,eff<∞M_3, ws,M_3, eff<∞ such that ⋆[|s~(Y,η⋆)|3∣X]≤M3,wsE [| s(Y,η )|^3 X]≤ M_3, ws and ⋆[|s(Y,η⋆)|3∣X]≤M3,effE [|s(Y,η )|^3 X]≤ M_3, eff for all admissible X; (R3) Tangent-projection bound: supx∈‖Px‖F≤Cprojd¯/d⋆ _x \|P_Tx\|_F≤ C_ proj d/d . F.1 Berry–Esseen bound for the leading i.i.d. term Proposition F.1 (Berry–Esseen for the whitened leading term). Let Zi:=ϕ~0⋆(Xi,Yi)=s~(Yi,ηi⋆)⟨Hws⋆,Xi⟩Z_i:= φ_0 (X_i,Y_i)= s(Y_i, _i )\, H _ ws,X_i with Hws⋆=d⋆PΓH _ ws=d P_T , and let vws2:=Var(Z1)v_ ws^2:=Var(Z_1). Under conditions (R1)–(R3) and the isotropic design assumption, there exists a constant C>0C>0, depending only on cI,CI,M3,wsc_I,C_I,M_3, ws, and CprojC_ proj, such that supt∈ℝ|P(1nvws∑i=1nZi≤t)−Φ(t)|≤Cd¯n. _t |P\! ( 1 n\,v_ ws _i=1^nZ_i≤ t )- (t) |\;≤\;C dn. Proof. We divide the argument into four steps. Step 1: mean zero. Because the whitened score s~(y,η)=s(y,η)/I(η) s(y,η)=s(y,η)/I(η) is conditionally centered, ⋆[s~(Y,η⋆)∣X]=0,E [ s(Y,η ) X]=0, we have ⋆Z1=⋆(⋆[s~(Y,η⋆)∣X]⟨Hws⋆,X⟩)=0.E Z_1=E \! (E [ s(Y,η ) X]\, H _ ws,X )=0. Hence Z1,…,ZnZ_1,…,Z_n are i.i.d. mean-zero random variables. Step 2: second moment. By definition, ⋆[s(Y,η⋆)2∣X]=I(η⋆)E [s(Y,η )^2 X]=I(η ), so ⋆[s~(Y,η⋆)2∣X]=⋆[s(Y,η⋆)2∣X]I(η⋆)2=1I(η⋆).E [ s(Y,η )^2 X]= E [s(Y,η )^2 X]I(η )^2= 1I(η ). Therefore, by the tower property, vws2=⋆[Z12]=⋆[s~(Y,η⋆)2⟨Hws⋆,X⟩2]=⋆[⟨Hws⋆,X⟩2I(η⋆)].v_ ws^2=E [Z_1^2]=E \! [ s(Y,η )^2\, H _ ws,X ^2 ]=E \! [ H _ ws,X ^2I(η ) ]. Using condition (R1) for the sandwich 1/CI≤1/I(η⋆)≤1/cI1/C_I≤ 1/I(η )≤ 1/c_I, together with the isotropic design identity ⋆⟨Hws⋆,X⟩2=‖Hws⋆‖F2/d⋆E H _ ws,X ^2=\|H _ ws\|_F^2/d (valid because Hws⋆∈H _ ws ): 1CI‖Hws⋆‖F2d⋆≤vws2≤1cI‖Hws⋆‖F2d⋆. 1C_I \|H _ ws\|_F^2d \;≤\;v_ ws^2\;≤\; 1c_I \|H _ ws\|_F^2d . (F.1) In particular, the cube of the standard deviation satisfies vws3≥CI−3/2‖Hws⋆‖F3(d⋆)3/2.v_ ws^3\;≥\;C_I^-3/2\, \|H _ ws\|_F^3(d )^3/2. (F.2) Step 3: third absolute moment. Starting from ⋆|Z1|3=⋆[|s~(Y,η⋆)|3|⟨Hws⋆,X⟩|3]E |Z_1|^3=E \! [| s(Y,η )|^3\,| H _ ws,X |^3 ], we condition on X and apply (R2): ⋆|Z1|3=⋆(⋆[|s~(Y,η⋆)|3∣X]|⟨Hws⋆,X⟩|3)≤M3,ws⋆|⟨Hws⋆,X⟩|3.E |Z_1|^3=E \! (E [| s(Y,η )|^3 X]\,| H _ ws,X |^3 )\;≤\;M_3, ws\,E | H _ ws,X |^3. (F.3) We now bound ⋆|⟨Hws⋆,X⟩|3E | H _ ws,X |^3 using the elementary inequality |W|3≤(sup|W|)|W|2,E|W|^3\;≤\; ( |W| )\,E|W|^2, (F.4) applied with W=⟨Hws⋆,X⟩W= H _ ws,X . Because Hws⋆∈H _ ws , the inner product satisfies ⟨Hws⋆,x⟩=⟨Hws⋆,Px⟩ H _ ws,x = H _ ws,P_Tx , so by Cauchy–Schwarz and condition (R3): supx∈|⟨Hws⋆,x⟩|≤‖Hws⋆‖Fsupx∈‖Px‖F≤Cproj‖Hws⋆‖Fd¯/d⋆. _x | H _ ws,x |\;≤\;\|H _ ws\|_F\, _x \|P_Tx\|_F\;≤\;C_ proj\,\|H _ ws\|_F\, d/d . (F.5) The isotropic design identity gives ⋆⟨Hws⋆,X⟩2=‖Hws⋆‖F2/d⋆E H _ ws,X ^2=\|H _ ws\|_F^2/d . Substituting (F.5) and this identity into (F.4): ⋆|⟨Hws⋆,X⟩|3≤Cproj‖Hws⋆‖F3d¯(d⋆)3/2.E | H _ ws,X |^3\;≤\;C_ proj\, \|H _ ws\|_F^3\, d(d )^3/2. (F.6) Combining (F.3) and (F.6): ⋆|Z1|3≤M3,wsCproj‖Hws⋆‖F3d¯(d⋆)3/2.E |Z_1|^3\;≤\;M_3, ws\,C_ proj\, \|H _ ws\|_F^3\, d(d )^3/2. (F.7) Step 4: standardized third moment and Berry–Esseen. Dividing (F.7) by the lower bound (F.2): ⋆|Z1|3vws3≤M3,wsCprojCI3/2d¯=:C0d¯. E |Z_1|^3v_ ws^3\;≤\;M_3, ws\,C_ proj\,C_I^3/2\, d\;=:\;C_0 d. (F.8) This is the key bound: the standardized third absolute moment is only O(d¯)O( d). Applying the classical Berry–Esseen theorem to the i.i.d. mean-zero variables Z1,…,ZnZ_1,…,Z_n: supt∈ℝ|P(1nvws∑i=1nZi≤t)−Φ(t)|≤CBEn⋆|Z1|3vws3≤CBEC0d¯n=Cd¯n, _t |P\! ( 1 n\,v_ ws _i=1^nZ_i≤ t )- (t) |\;≤\; C_ BE n\, E |Z_1|^3v_ ws^3\;≤\; C_ BE\,C_0 d n=C dn, (F.9) where C=CBEC0=CBEM3,wsCprojCI3/2C=C_ BE\,C_0=C_ BE\,M_3, ws\,C_ proj\,C_I^3/2 and CBE≤0.4748C_ BE≤ 0.4748 is the universal Berry–Esseen constant. ∎ F.2 Tangent-projection bound for pairwise atoms We verify that condition (R3) follows from the standard basis-tensor projection bound. Lemma F.2 (Pairwise tangent-projection bound). Suppose that for every canonical basis tensor EωE_ω, ‖PEω‖F≤C0d¯/d⋆.\|P_TE_ω\|_F\;≤\;C_0 d/d . Then for every pairwise-comparison design tensor X=Eω+−Eω−X=E_ω^+-E_ω^-, ‖PX‖F≤ 2C0d¯/d⋆.\|P_TX\|_F\;≤\;2C_0 d/d . Hence condition (R3) holds with Cproj=2C0C_ proj=2C_0. Proof. By linearity of P_T and the triangle inequality, ‖PX‖F=‖PEω+−PEω−‖F≤‖PEω+‖F+‖PEω−‖F≤ 2C0d¯/d⋆.\|P_TX\|_F=\|P_TE_ω^+-P_TE_ω^-\|_F\;≤\;\|P_TE_ω^+\|_F+\|P_TE_ω^-\|_F\;≤\;2C_0 d/d . The basis-tensor bound ‖PEω‖F≤C0d¯/d⋆\|P_TE_ω\|_F≤ C_0 d/d is the standard μ-incoherence bound from the tensor-completion literature, and holds with C0=C(μ,r,m)C_0=C(μ,r,m). The pairwise design introduces only an absolute constant factor 22. ∎ F.3 Berry–Esseen bound for the efficient leading term The same argument applies, with minimal modification, to the efficient (non-whitened) leading term. Proposition F.3 (Berry–Esseen for the efficient leading term). Let Zieff:=ϕ⋆(Xi,Yi)=s(Yi,ηi⋆)⟨H⋆,Xi⟩Z_i eff:=φ (X_i,Y_i)=s(Y_i, _i )\, H ,X_i with H⋆=A−1PΓH =A^-1P_T , and let veff2:=Var(Z1eff)v_ eff^2:=Var(Z_1 eff). Under conditions (R1)–(R3) and the isotropic design assumption, supt∈ℝ|P(1nveff∑i=1nZieff≤t)−Φ(t)|≤Cd¯n. _t |P\! ( 1 n\,v_ eff _i=1^nZ_i eff≤ t )- (t) |\;≤\;C dn. Proof. The proof follows the same four-step structure as Proposition F.1; we record only the changes. Step 1. Mean zero follows from ⋆[s(Y,η⋆)∣X]=0E [s(Y,η ) X]=0. Step 2. Since ⋆[s(Y,η⋆)2∣X]=I(η⋆)E [s(Y,η )^2 X]=I(η ), the second moment is veff2=⋆[I(η⋆)⟨H⋆,X⟩2].v_ eff^2=E \! [I(η )\, H ,X ^2 ]. Using condition (R1) and the isotropic design: cId⋆‖H⋆‖F2≤veff2≤CId⋆‖H⋆‖F2. c_Id \|H \|_F^2\;≤\;v_ eff^2\;≤\; C_Id \|H \|_F^2. (F.10) Step 3. Using condition (R2), ⋆[|s|3∣X]≤M3,effE [|s|^3 X]≤ M_3, eff, and the same decomposition |W|3≤(sup|W|)|W|2E|W|^3≤( |W|)E|W|^2: ⋆|Z1eff|3≤M3,effCproj‖H⋆‖F3d¯(d⋆)3/2.E |Z_1 eff|^3\;≤\;M_3, eff\,C_ proj\, \|H \|_F^3\, d(d )^3/2. (F.11) Step 4. Dividing by veff3≥cI3/2‖H⋆‖F3/(d⋆)3/2v_ eff^3≥ c_I^3/2\|H \|_F^3/(d )^3/2 (from (F.10)): ⋆|Z1eff|3veff3≤M3,effCprojcI3/2d¯, E |Z_1 eff|^3v_ eff^3\;≤\; M_3, eff\,C_ projc_I^3/2\, d, and the Berry–Esseen theorem gives the stated bound. The ‖H⋆‖F\|H \|_F terms cancel exactly as before, so the leading-term Berry–Esseen rate is the same O(d¯/n)O( d/n) for both the whitened and efficient estimators. ∎ F.4 Combined Berry–Esseen bound with remainder The one-step decomposition ψ^n−ψ(T⋆)=1n∑i=1nZi+Rn ψ_n-ψ(T )= 1n _i=1^nZ_i+R_n contains both the i.i.d. leading term and the remainder RnR_n. The leading-term Berry–Esseen bounds of Propositions F.1 and F.3 quantify the Gaussian approximation error of the first part. We now derive the full Berry–Esseen bound by carefully incorporating the remainder, which is random and dependent on the same evaluation-fold data. Theorem F.4 (Combined Berry–Esseen bound). Let Tn:=n(ψ^n−ψ(T⋆))/vT_n:= n( ψ_n-ψ(T ))/v denote the standardized one-step estimator, where v=vwsv=v_ ws or veffv_ eff as appropriate. Write Tn=Sn+ρnT_n=S_n+ _n where Sn:=1nv∑i=1nZiandρn:=nRnvS_n:= 1 n\,v _i=1^nZ_i _n:= n\,R_nv are the standardized leading term and standardized remainder, respectively. Let rn>0r_n>0 be any deterministic bound and define the event ℰn:=|ρn|≤rnE_n:=\| _n|≤ r_n\. Then supt∈ℝ|P(Tn≤t)−Φ(t)|≤ΔBE+rn2π+P(ℰnc), _t \! |P\! (T_n≤ t )- (t) |\;≤\; _ BE+ r_n 2π+P(E_n^c), (F.12) where ΔBE=Cd¯/n _ BE=C d/n is the leading-term Berry–Esseen error from Proposition F.1 (resp. F.3). Proof. Fix any t∈ℝt . Since Tn=Sn+ρnT_n=S_n+ _n, we decompose the probability according to the event ℰnE_n. Upper bound. On the event ℰnE_n, |ρn|≤rn| _n|≤ r_n, so Tn≤tT_n≤ t implies Sn=Tn−ρn≤t+rnS_n=T_n- _n≤ t+r_n. Therefore P(Tn≤t,ℰn)≤P(Sn≤t+rn).P(T_n≤ t,\;E_n)\;≤\;P(S_n≤ t+r_n). Adding the failure probability: P(Tn≤t) P(T_n≤ t) =P(Tn≤t,ℰn)+P(Tn≤t,ℰnc) \;=\;P(T_n≤ t,\;E_n)+P(T_n≤ t,\;E_n^c) ≤P(Sn≤t+rn)+P(ℰnc). \;≤\;P(S_n≤ t+r_n)+P(E_n^c). (F.13) Now apply the Berry–Esseen bound from Proposition F.1 (or F.3), which holds uniformly in u: P(Sn≤t+rn)≤Φ(t+rn)+ΔBE.P(S_n≤ t+r_n)\;≤\; (t+r_n)+ _ BE. Since Φ is Lipschitz with constant suptΦ′(t)=suptϕ(t)=1/2π _t (t)= _tφ(t)=1/ 2π, Φ(t+rn)≤Φ(t)+rn2π. (t+r_n)\;≤\; (t)+ r_n 2π. Substituting into (F.13): P(Tn≤t)≤Φ(t)+ΔBE+rn2π+P(ℰnc).P(T_n≤ t)\;≤\; (t)+ _ BE+ r_n 2π+P(E_n^c). (F.14) Lower bound. On ℰnE_n, if Sn≤t−rnS_n≤ t-r_n then Tn=Sn+ρn≤(t−rn)+rn=tT_n=S_n+ _n≤(t-r_n)+r_n=t. Therefore P(Sn≤t−rn,ℰn)≤P(Tn≤t).P(S_n≤ t-r_n,\;E_n)\;≤\;P(T_n≤ t). Since P(Sn≤t−rn,ℰn)≥P(Sn≤t−rn)−P(ℰnc)P(S_n≤ t-r_n,\;E_n)≥ P(S_n≤ t-r_n)-P(E_n^c): P(Tn≤t) P(T_n≤ t) ≥P(Sn≤t−rn)−P(ℰnc) \;≥\;P(S_n≤ t-r_n)-P(E_n^c) ≥Φ(t−rn)−ΔBE−P(ℰnc) \;≥\; (t-r_n)- _ BE-P(E_n^c) ≥Φ(t)−rn2π−ΔBE−P(ℰnc). \;≥\; (t)- r_n 2π- _ BE-P(E_n^c). (F.15) Conclusion. Combining (F.14) and (F.15): |P(Tn≤t)−Φ(t)|≤ΔBE+rn2π+P(ℰnc). |P(T_n≤ t)- (t) |\;≤\; _ BE+ r_n 2π+P(E_n^c). Taking the supremum over t∈ℝt gives (F.12). ∎ We now specialize to each estimator. Whitened estimator. By Theorem 5.2, with probability at least 1−d¯−c1- d^-c, |Rn|≤C‖Γ‖1d¯logc⁡d¯n.|R_n|\;≤\;C\,\| \|_1\, d ^c dn. The second-moment bound (F.1) gives vws≥CI−1/2‖Hws⋆‖F/d⋆≍d¯‖Γ‖1v_ ws≥ C_I^-1/2\|H _ ws\|_F/ d d\,\| \|_1. Therefore the standardized remainder satisfies, on the event ℰnE_n, |ρn|=n|Rn|vws≤C′n‖Γ‖1d¯logc⁡d¯/nd¯‖Γ‖1=C′d¯log2c⁡d¯n=:rn,| _n|= n\,|R_n|v_ ws\;≤\;C \, n\,\| \|_1\, d ^c d/n d\,\| \|_1=C d ^2c dn\;=:\;r_n, with P(ℰnc)≤d¯−cP(E_n^c)≤ d^-c. Substituting into (F.12) with ΔBE=Cd¯/n _ BE=C d/n from Proposition F.1: supt∈ℝ|P(n(ψ^nws−ψ(T⋆))vws≤t)−Φ(t)| _t \! |P\! ( n( ψ_n ws-ψ(T ))v_ ws≤ t )- (t) | (F.16) ≤Cd¯n⏟leading-term BE+C′2πd¯log2c⁡d¯n⏟remainder contribution+d¯−c⏟failure prob.=O(d¯log2c⁡d¯n). \;≤\; C dn_leading-term BE\;+\; C 2π d ^2c dn_remainder contribution\;+\; d^-c_failure prob.\;=\;O\! ( d ^2c dn ). This vanishes under the sample-size condition n≫d¯log2c⁡d¯n d ^2c d, which is the same condition required for the CLT. Since c>0c>0 can be taken arbitrarily large, the failure probability d¯−c d^-c is negligible compared to the other two terms. Efficient estimator. By Theorem 4.4, with probability at least 1−d¯−c1- d^-c, |Rn|≤CCA‖Γ‖1d¯logc⁡d¯n.|R_n|\;≤\;C\,C_A\,\| \|_1\, d ^c dn. Since veff≍d¯‖Γ‖1v_ eff d\,\| \|_1 by (F.10), the standardized remainder satisfies |ρn|≤CA⋅C′d¯log2c⁡d¯n=:rn.| _n|\;≤\;C_A· C d ^2c dn\;=:\;r_n. Substituting with ΔBE=Cd¯/n _ BE=C d/n from Proposition F.3: supt∈ℝ|P(n(ψ^neff−ψ(T⋆))veff≤t)−Φ(t)|≤O(CAd¯log2c⁡d¯n), _t \! |P\! ( n( ψ_n eff-ψ(T ))v_ eff≤ t )- (t) |\;≤\;O\! (C_A d ^2c dn ), (F.17) which vanishes under the CLT condition CA2d¯log2c⁡d¯/n→0C_A^2 d ^2c d/n→ 0, i.e., (4.11). Remark F.5 (Leading term vs. remainder). For both estimators, the leading-term Berry–Esseen error is O(d¯/n)O( d/n), independent of CAC_A. The difference between the whitened and efficient estimators lies entirely in the remainder: the whitened remainder has no CAC_A factor, while the efficient remainder carries the factor CAC_A, reflecting the additional estimation error from inverting the information operator. Appendix G Application to Tensor Completion Although this paper focuses on pairwise comparisons and the new challenges they introduce (identification constraints, non-constant Fisher information, and the non-commutativity bottleneck), the semiparametric framework developed here applies to general sampling mechanisms, observation patterns, and inference targets. To illustrate this point, we show in this section how the framework recovers and extends classical tensor completion results. Below we first outline the key steps. Lower bounds and efficiency gaps in existing work. The semiparametric efficiency bound Veff(ψ)=⟨PΓ,(PGP)−1PΓ⟩V_eff(ψ)= P_T ,(P_TGP_T)^-1P_T reveals that existing debiased estimators in the tensor completion literature—under heteroscedastic, non-additive noise, or non-uniform sampling—are not semiparametrically efficient. The reason is that these estimators are all special cases of the score-whitening method, which replaces the information operator G by a scalar multiple of the identity. This achieves the whitened variance Vws≥VeffV_ws≥ V_eff, with strict inequality whenever the Fisher information I(η⋆)I(η ) varies across entries. Whitening methods as natural adaptation. For additive sub-exponential noise with heteroscedastic variance, the natural residual score s(y,η)=y−ηs(y,η)=y-η automatically satisfies the whitening property with I(η)≡1I(η)≡ 1, so the score-whitened estimator is in fact fully efficient—and this is precisely the estimator used throughout the existing literature. For non-uniform sampling, IPW adjustment restores isotropy of the information operator, again reducing G to a scalar multiple of the identity. For 1-bit observations (e.g., logistic link), the score-whitening method of Section 5 applies identically to the pairwise case, yielding the first general CLT for debiased inference in 1-bit tensor completion. Efficient estimation beyond whitening. When the noise variance or sampling distribution is known (or can be estimated consistently), one can instead solve the full information equation PGPH^=PΓP_TGP_T H=P_T with the true G operator and construct the efficient one-step estimator that achieves VeffV_eff, strictly improving upon whitening whenever the information geometry is heterogeneous. The remaining analysis is identical to the pairwise comparison case, since these observation and sampling patterns affect only the information operator G, and the entire asymptotic theory of Sections 4–5 is stated in terms of G abstractly. We note that for additive noise with unknown variance, the information operator G involves the unknown noise variances and is harder to estimate reliably; in this case, the whitening approach is preferred. For general observation models such as tensor completion with additive noise, we assume the following signal strength and noise conditions: Assumption G.1 (Signal strength and sample size—general case). The Frobenius norm and the sample size satisfy ‖T⋆‖F≥cσd⋆\|T \|_F≥ cσ d (G.1) for an absolute constant c>0c>0, and n≥C0d¯logc⁡d¯n≥ C_0\, d ^c d (G.2) for a sufficiently large constant C0C_0 depending on μ,κ,r,mμ,κ,r,m. Assumption G.2 (Score regularity—general case). The score function sη(y,η):=∂ηℓ(y,η)s_η(y,η):= _η (y,η) satisfies: (i) centering: ⋆[sη(Y,η⋆)∣X]=0E [s_η(Y,η ) X]=0; (i) Fisher bounds: cI≤I(η⋆)≤CIc_I≤ I(η )≤ C_I a.s.; (i) derivative bounds: |s˙η(y,η)|≤B1| s_η(y,η)|≤ B_1, |s¨η(y,η)|≤B2| s_η(y,η)|≤ B_2; (iv) tail control: sη(Y,η^)s_η(Y, η) is conditionally sub-exponential with bounded scale parameter. In the pairwise-comparison (and 1-bit) setting, both assumptions are automatically satisfied under the bounded signal condition ‖T⋆‖∞≤B\|T \|_∞≤ B (see Section 4.2). In this section we specialize our general framework to entrywise matrix/tensor completion. We first derive the diagonal Gram (Fisher) operator and state the semiparametric efficiency bound. We then show that existing debiased estimators in the literature are special cases of the score-whitening method and identify when they do and do not achieve full efficiency. For observation models where the information operator can be estimated (1-bit observations, additive noise with known variance), we construct the semiparametrically efficient one-step estimator that strictly improves upon whitening. Finally, we show how score whitening handles heteroscedastic noise, Bernoulli observations, and non-uniform sampling at the optimal sample complexity. G.1 Entrywise observation model and the Gram operator We specialize to the matrix case ℍ=ℝd1×d2H=R^d_1× d_2 with the Frobenius inner product and canonical basis Ejk1≤j≤d1, 1≤k≤d2\E_jk\_1≤ j≤ d_1,\,1≤ k≤ d_2, where [Ejk]ab=(a,b)=(j,k)[E_jk]_ab=1\(a,b)=(j,k)\. In entrywise tensor completion, each sample i selects an entry (ji,ki)(j_i,k_i) from a sampling distribution on [d1]×[d2][d_1]×[d_2] with pjk:=ℙ((ji,ki)=(j,k)),p=d1d2,p_jk:=P ((j_i,k_i)=(j,k) ), p=d_1d_2, and we set the design X(i)=EjikiX^(i)=E_j_ik_i. Conditional on X(i)X^(i), the response Y(i)Y^(i) depends on T⋆T only through the scalar parameter η(i)=⟨T⋆,X(i)⟩=Tjiki⋆η^(i)= T ,X^(i) =T _j_ik_i, with a scalar log-likelihood ℓ(y,η) (y,η). The per-entry score and Fisher information are sη(y,η):=∂ηℓ(y,η),I(η):=[sη(Y,η)2|η].s_η(y,η):= ∂η (y,η), I(η):=E [s_η(Y,η)^2\, |\,η ]. Diagonal Gram operator. Since X=EjkX=E_jk with probability pjkp_jk, we have Xjk2=Xjk∈0,1X_jk^2=X_jk∈\0,1\ and the Gram operator acts diagonally: G(H)=∑j,kλjkHjkEjk,λjk:=pjkI(Tjk⋆).G(H)= _j,k _jk\,H_jk\,E_jk, _jk:=p_jk\,I (T _jk ). (G.3) Each weight λjk _jk is the product of two factors: the sampling probability pjkp_jk and the Fisher information I(Tjk⋆)I(T _jk) at the true entry value. The Gram norm is ‖H‖G2=∑j,kλjkHjk2\|H\|_G^2= _j,k _jkH_jk^2. Efficiency bound. The full-model information equation G(HΓfull)=ΓG(H_ ^full)= is solved entrywise: (HΓfull)jk=Γjk/λjk(H_ ^full)_jk= _jk/ _jk. In the low-rank model, as in the pairwise case (Section 3), we define the restricted information operator A:=P∘G∘PA:=P_T G P_T, and the semiparametric efficiency bound for ψ(T)=⟨Γ,T⟩Fψ(T)= ,T _F is Veff(ψ)=⟨P(Γ),A−1P(Γ)⟩F.V_eff(ψ)= P_T( ),\,A^-1P_T( ) _F. (G.4) G.2 When G is close to the identity When the weights λjk _jk are approximately constant—for instance, under uniform sampling (pjk=1/p_jk=1/p) with homoscedastic noise (I(η)≡cI(η)≡ c)—the Gram operator satisfies G=(c/p)IdG=(c/p)\,Id and trivially commutes with P_T. In this regime, the general one-step estimator of Section 4 achieves the semiparametric efficiency bound at the optimal sample complexity n≳d¯polylog(d¯)n d\,polylog( d). More generally, two routes to full efficiency are available (cf. Section 4): (a) If the sampling probabilities pjk\p_jk\ and Fisher weights I(Tjk⋆)\I(T _jk)\ satisfy maxjk⁡λjk/minjk⁡λjk≤B0(μ,r,κ) _jk _jk/ _jk _jk≤ B_0(μ,r,κ) for a dimension-independent constant B0B_0, then G is a bounded perturbation of the identity. The constant CAC_A in (4.16) remains O(1)O(1), and the one-step estimator achieves the bound (G.4) at the statistically optimal sample complexity. (b) Even with substantial heterogeneity, the sample complexity n≥Cd¯2(log⁡d¯)n≥ C d^2\,( d) absorbs the non-commutativity cost CAC_A, so the general estimator again achieves full efficiency. Remark G.3 (Uniform sampling with Gaussian noise). Under uniform sampling pjk=1/p_jk=1/p and Gaussian noise with variance σ2σ^2, we have sη(y,η)=(y−η)/σ2s_η(y,η)=(y-η)/σ^2, I(η)=1/σ2I(η)=1/σ^2, and λjk=1/(σ2p) _jk=1/(σ^2p). The Gram operator is G=(1/σ2p)IdG=(1/σ^2p)\,Id, the efficiency bound reduces to VeffMC=σ2p‖P(Γ)‖F2V_eff^MC=σ^2p\,\|P_T( )\|_F^2, and the efficient debiased estimator is ψ^MC=⟨Γ,T^⟩+1n∑i=1np⟨P(Γ),Xi⟩(Yi−⟨T^,Xi⟩), ψ_MC= , T + 1n _i=1^np\, P_T( ),X_i \,(Y_i- T,X_i ), recovering existing results in the matrix completion literature. G.3 Semiparametric efficiency bound and gaps in existing literature We now show that the semiparametric efficiency bound from our general framework reveals a precise efficiency gap in existing tensor completion inference methods under heterogeneous settings. G.3.1 The efficiency bound The semiparametric efficiency bound for the linear functional ψ(T)=⟨Γ,T⟩ψ(T)= ,T is Veff(ψ)=⟨P(Γ),A−1P(Γ)⟩F,A=P∘G∘P.V_eff(ψ)= P_T( ),\,A^-1P_T( ) _F, A=P_T G P_T. Since G is diagonal with weights λjk=pjkI(Tjk⋆) _jk=p_jkI(T _jk) (cf. (G.3)), solving the full-model information equation gives (HΓfull)jk=Γjk/λjk(H_ ^full)_jk= _jk/ _jk. In the low-rank model, the efficient direction H⋆=A−1P(Γ)H =A^-1P_T( ) additionally incorporates the tangent-space projection, and the resulting variance VeffV_eff is strictly smaller than any variance achievable by estimators that do not solve this operator equation. G.3.2 Why existing debiased estimators are not efficient Existing debiased estimators for tensor completion, including those of Ma and Xia (2024), are all based on the residual score s(y,η)=y−ηs(y,η)=y-η (for additive noise) or the whitened score s~=s/I(η) s=s/I(η) (for non-additive noise). In both cases, the effective Gram operator reduces to a scalar multiple of the identity: G0=cIdG_0=c\,Id on the tangent space. The oracle direction becomes Hws⋆=c−1PΓH_ ws =c^-1P_T , and the resulting asymptotic variance is Vws=⋆[⟨Hws⋆,X⟩2I(η⋆)].V_ ws=E \! [ H_ ws ,X ^2I(η ) ]. By the Cauchy–Schwarz inequality and the definition of VeffV_eff, one always has Vws≥VeffV_ ws≥ V_eff, with equality if and only if I(η⋆)I(η ) is constant across all observed entries. In the following cases, the inequality is strict: • Heteroscedastic additive noise: When the noise variance σjk2σ^2_jk varies across entries, the whitened score s=y−ηs=y-η has I(η)≡1I(η)≡ 1, so Vws=VeffV_ ws=V_eff—the whitening approach is automatically efficient. • 1-bit observations (e.g., logistic link): The Fisher information I(η)=f′(η)2/[f(η)(1−f(η))]I(η)=f (η)^2/[f(η)(1-f(η))] varies with η=Tjk⋆η=T _jk. Here Vws>VeffV_ ws>V_eff whenever the entries of T⋆T are not all identical. • Non-uniform sampling with additive noise: If one applies IPW whitening, the effective variance depends on both the sampling probabilities and the noise variances. The whitened estimator adapts to sampling heterogeneity but may not achieve the true efficiency bound. G.3.3 Constructing the efficient estimator When the information operator G can be estimated consistently, one can construct the semiparametrically efficient one-step estimator that achieves VeffV_eff. G.3.4 Additive noise with known (or estimable) variance Under the additive model Y=Tjk⋆+εY=T _jk+ with Var(ε∣X=Ejk)=σjk2Var( X=E_jk)=σ^2_jk, the score is s(y,η)=y−ηs(y,η)=y-η and the Fisher information at entry (j,k)(j,k) is Ijk=1/σjk2I_jk=1/σ^2_jk. With sampling probability pjkp_jk, the Gram operator acts diagonally: G(H)jk=pjkσjk2Hjk.G(H)_jk= p_jkσ^2_jk\,H_jk. (G.5) The weights λjk=pjk/σjk2 _jk=p_jk/σ^2_jk combine both the sampling intensity and the signal-to-noise ratio at each entry. The Gram norm is ‖H‖G2=∑j,k(pjk/σjk2)Hjk2\|H\|_G^2= _j,k(p_jk/σ^2_jk)\,H_jk^2. Efficiency bound. The restricted information operator is A=P∘G∘PA=P_T G P_T, and the semiparametric efficiency bound is Veff(ψ)=⟨P(Γ),A−1P(Γ)⟩F.V_eff(ψ)= P_T( ),\,A^-1P_T( ) _F. Under uniform sampling (pjk=1/p_jk=1/p) and homoscedastic noise (σjk2≡σ2σ^2_jk≡σ^2), this simplifies to Veff=σ2p‖P(Γ)‖F2V_eff=σ^2p\,\|P_T( )\|_F^2, recovering the standard result (Remark G.3). The efficient one-step estimator. When the variances σjk2σ^2_jk are known, we construct G^(H)jk=(pjk/σjk2)Hjk G(H)_jk=(p_jk/σ^2_jk)\,H_jk and compute ψ^eff=⟨Γ,T^⟩+1n∑i=1n(Yi−⟨T^,Xi⟩)⟨H^eff,Xi⟩,H^eff=(P^G^P^)−1P^Γ. ψ_ eff= , T + 1n _i=1^n(Y_i- T,X_i )\, H_ eff,X_i , H_ eff=( P_T G P_T)^-1 P_T . (G.6) The analysis follows the proof of Theorem 4.5 exactly: the score is s=Y−η^s=Y- η, the score perturbation is s(Y,η^)−s(Y,η⋆)=−⟨Δ,X⟩s(Y, η)-s(Y,η )=- ,X with no Taylor remainder (since s is affine in η), and the H-direction bias is −⟨Δ,G(H^eff−Heff⋆)⟩- ,G( H_ eff-H_ eff ) . The resolvent perturbation argument (Appendix C.10) handles the error from P^≠P P_T≠ P_T in the operator inversion. The asymptotic variance is Veff=⟨PΓ,A−1PΓ⟩V_eff= P_T ,A^-1P_T . Under heteroscedastic noise, Veff<VwsV_eff<V_ ws strictly. To see this, note that the whitened estimator uses Hws⋆=pPΓH_ ws =p\,P_T (under uniform sampling) with variance Vws=p∑jkσjk2(PΓ)jk2V_ ws=p _jkσ^2_jk\,(P_T )_jk^2, while the efficient estimator adapts its direction entry by entry via G, placing more weight on entries with smaller noise. When noise variances are unknown. When σjk2σ^2_jk is unknown, estimating G requires estimating each σjk2σ^2_jk, which is difficult without repeated observations at the same entry. In this case, the whitening approach (Section G.4.1) is preferred: the residual score s=Y−ηs=Y-η automatically has effective Fisher information I≡1I≡ 1, making the whitened estimator fully efficient without knowing the noise distribution. This is why the residual-based debiased estimators used throughout the existing literature are in fact optimal for additive noise. G.3.5 1-bit observations with known link Under the 1-bit model Y∈0,1Y∈\0,1\ with ℙ(Y=1∣η)=f(η)P(Y=1 η)=f(η) for a known link f (e.g., logistic f(η)=(1+e−η)−1f(η)=(1+e^-η)^-1) and uniform sampling pjk=1/p_jk=1/p, the score and Fisher information are s(y,η)=(y−f(η))f′(η)f(η)(1−f(η)),I(η)=f′(η)2f(η)(1−f(η)).s(y,η)= (y-f(η))\,f (η)f(η)(1-f(η)), I(η)= f (η)^2f(η)(1-f(η)). The Gram operator is diagonal with entry-dependent weights: G(H)jk=1pI(Tjk⋆)Hjk.G(H)_jk= 1p\,I(T _jk)\,H_jk. (G.7) Unlike the additive noise case, the Fisher information I(Tjk⋆)I(T _jk) varies with the latent entry value, making the Gram operator genuinely non-isotropic. The efficient one-step estimator. Since the link function f is known and T⋆T can be estimated via T T, one can construct a consistent plug-in estimator of G: G^(H)jk=1pI(T^jk)Hjk. G(H)_jk= 1p\,I( T_jk)\,H_jk. The entrywise error is controlled by the entrywise accuracy of T T: under the bounded-signal assumption ‖T⋆‖∞≤B\|T \|_∞≤ B and Lipschitz continuity of I(⋅)I(·), |λ^jk−λjk|=1p|I(T^jk)−I(Tjk⋆)|≤LIp|T^jk−Tjk⋆|,| λ_jk- _jk|= 1p|I( T_jk)-I(T _jk)|≤ L_Ip\,| T_jk-T _jk|, where LIL_I is the Lipschitz constant of I on [−B−1,B+1][-B-1,B+1]. Thus the entrywise guarantee ‖T^−T⋆‖∞=o(1)\| T-T \|_∞=o(1) (already required for the one-step estimator) ensures consistency of G G. The efficient one-step estimator for 1-bit completion is ψ^eff=⟨Γ,T^⟩+1n∑i=1ns(Yi,⟨T^,Xi⟩)⟨H^eff,Xi⟩,H^eff=(P^G^P^)−1P^Γ. ψ_ eff= , T + 1n _i=1^ns(Y_i, T,X_i )\, H_ eff,X_i , H_ eff=( P_T G P_T)^-1 P_T . (G.8) The remaining analysis follows the proof of Theorem 4.5 verbatim: the additional error from G^≠G G≠ G is handled by the resolvent perturbation argument (Appendix C.10), and the CLT holds under the same sample-complexity conditions. Asymptotic variance. The efficient variance is Veff=⟨P(Γ),A−1P(Γ)⟩F=p∑j,k[P(Γ)]jk2I(Tjk⋆)⋅[A−1P(Γ)]jk[P(Γ)]jk/I(Tjk⋆),V_eff= P_T( ),\,A^-1P_T( ) _F=p _j,k [P_T( )]_jk^2I(T _jk)· [A^-1P_T( )]_jk[P_T( )]_jk/I(T _jk), which is strictly smaller than the whitened variance Vws=p∑jk[P(Γ)]jk2/I(Tjk⋆)V_ ws=p _jk[P_T( )]_jk^2/I(T _jk) whenever the Fisher information I(Tjk⋆)I(T _jk) varies across entries. Intuitively, the efficient estimator optimally reweights contributions from each entry according to both the tangent-space geometry and the local Fisher information, whereas the whitening approach treats all entries equally after normalization. Comparison with whitening. The whitened estimator for 1-bit observations (Section G.4.2) uses s~(y,η)=s(y,η)/I(η) s(y,η)=s(y,η)/I(η) and the simplified direction H^ws=pP^Γ H_ ws=p\, P_T . It achieves Vws=p∑jk[P(Γ)]jk2/I(Tjk⋆)≥VeffV_ ws=p _jk[P_T( )]_jk^2/I(T _jk)≥ V_eff. The gap Vws−VeffV_ ws-V_eff grows with the heterogeneity of I(Tjk⋆)I(T _jk) across entries. However, the whitening approach has the practical advantage of not requiring operator inversion (P^G^P P_T G P_T), making it more robust in small-sample regimes. This is the first CLT for debiased inference in 1-bit tensor completion under general link functions. G.4 Score whitening for tensor completion When G departs substantially from the identity and the sample size is not large enough for route (b) above, the score-whitening method of Section 5 provides valid inference at the optimal sample complexity, albeit with a potentially larger variance. We treat two important observation models. G.4.1 Additive noise Suppose the observation model is Y=Tjk⋆+εY=T _jk+ , where ε has mean zero and possibly entry-dependent variance σjk2σ^2_jk; the noise distribution may be sub-Gaussian, sub-exponential, or otherwise—crucially, we do not require knowledge of σjk2σ^2_jk or even of the noise family. The key observation is that the score for additive noise is simply s(y,η)=y−η,s(y,η)=y-η, regardless of the noise distribution. The whitened score of Section 5 is s~(y,η)=s(y,η)/I(η) s(y,η)=s(y,η)/I(η). For Gaussian noise with variance σjk2σ^2_jk, the “true” efficient score is (Y−η)/σjk2(Y-η)/σ^2_jk and the Fisher information is I(η)=1/σjk2I(η)=1/σ^2_jk; dividing the two gives s~=Y−η s=Y-η, i.e., the unknown variance cancels. More generally, for any additive noise model one can use the score s=Y−ηs=Y-η directly. Since [s2∣η]=σjk2E[s^2 η]=σ^2_jk and [∂ηs∣η]=−1E[ _ηs η]=-1, the whitened score satisfies properties (i)–(i) of Section 5.1 with I(η)≡1I(η)≡ 1 in the “natural” parameterization. The resulting Gram operator under score whitening and uniform sampling is G0(H)=1pH,A0=1pP,A0−1=pP.G_0(H)= 1p\,H, A_0= 1p\,P_T, A_0^-1=p\,P_T. The score-whitened one-step estimator takes the form ψ^ws=⟨Γ,T^⟩+1n∑i=1n(Yi−⟨T^,Xi⟩)⟨H^ws,Xi⟩,H^ws=pP^Γ. ψ_ ws= , T + 1n _i=1^n(Y_i- T,X_i )\, H_ ws,X_i , H_ ws=p\, P_T . (G.9) Since I(η)≡1I(η)≡ 1 under the natural score, the whitened estimator is fully semiparametrically efficient: Vws=VeffV_ ws=V_ eff, regardless of heteroscedasticity (Theorem 5.2). Under homoscedastic noise (σjk2≡σ2σ^2_jk≡σ^2), the variance simplifies to Vws=σ2p‖P(Γ)‖F2V_ ws=σ^2p\,\|P_T( )\|_F^2. Remark G.4 (Why the noise distribution is irrelevant). The estimator (G.9) uses only the residual Yi−⟨T^,Xi⟩Y_i- T,X_i as the score, requiring no knowledge of the noise variance, distribution family, or heteroscedastic structure. This is a distinctive advantage of the score-whitening approach for additive noise models: full semiparametric efficiency is achieved with a single, distribution-free estimator. G.4.2 Bernoulli / 1-bit observations In 1-bit matrix completion, Y∈0,1Y∈\0,1\ with ℙ(Y=1∣η)=f(η)P(Y=1 η)=f(η) for some link function f (e.g., logistic f(η)=(1+e−η)−1f(η)=(1+e^-η)^-1). The score is s(y,η)=(y−f(η))f′(η)/(f(η)(1−f(η)))s(y,η)=(y-f(η))\,f (η)/ (f(η)(1-f(η)) ) and the Fisher information is I(η)=f′(η)2f(η)(1−f(η)).I(η)= f (η)^2f(η)(1-f(η)). The Fisher information varies with η=Tjk⋆η=T _jk, so the Gram operator (G.3) is genuinely non-isotropic. The whitened score s~(y,η)=s(y,η)/I(η) s(y,η)=s(y,η)/I(η) depends on η only, which we estimate via T T. Since I(η)I(η) is a known function of the estimable parameter η, the score-whitening method of Section 5 applies with the same analysis. The score-whitened one-step estimator is ψ^ws=⟨Γ,T^⟩+1n∑i=1ns~(Yi,⟨T^,Xi⟩)⟨H^ws,Xi⟩,H^ws=pP^Γ. ψ_ ws= , T + 1n _i=1^n s(Y_i, T,X_i )\, H_ ws,X_i , H_ ws=p\, P_T . (G.10) The asymptotic variance is Vws=⋆[⟨pPΓ,X⟩2I(η⋆)]=p∑j,kpjk[P(Γ)]jk2I(Tjk⋆).V_ ws=E \! [ p\,P_T ,X ^2I(η ) ]=p _j,k p_jk\, [P_T( ) ]_jk^2I(T _jk). In general Vws≥VeffV_ ws≥ V_ eff, with equality when I(η⋆)I(η ) is constant across all entries (i.e., when f is nearly linear or all entries of T⋆T are close). The CLT and remainder bounds follow from Theorem 5.2. G.5 Non-uniform sampling When the sampling probabilities pjk\p_jk\ are non-uniform, the inverse-probability weighting (IPW) framework of Section 5.2 restores isotropy. Specifically, choosing the uniform reference distribution qjk=1/pq_jk=1/p and defining weights wjk=qjk/pjkw_jk=q_jk/p_jk, the IPW-whitened estimator from (5.14) applies directly. The effective Gram operator under the IPW-whitened score becomes (1/p)Id(1/p)\,Id, exactly as in the uniform case. The factorization λjk=pjk⋅I(Tjk⋆) _jk=p_jk· I(T _jk) in (G.3) makes the decomposition transparent: score whitening removes the Fisher information heterogeneity (Section G.4), while IPW removes the sampling heterogeneity (Section 5.2). The two adjustments compose naturally. When pjk\p_jk\ are unknown, a consistent estimator p^jk p_jk is needed. As discussed in Section 5.2, if the sampling mechanism admits a low-rank or structured form, the weights can be estimated at the parametric rate with only second-order impact on the inference; see the analysis following (5.16). Appendix H Entrywise Refinement and Convex Initialization This section contains the detailed treatment of the entrywise refinement procedure and the convex initialization discussed in Assumption 4.1. We present three main results: (i) a row-wise refinement for matrices observed through pairwise comparisons (Appendix H.1), (i) the extension to general-order tensors (Appendix H.2), and (i) a nuclear-norm penalized convex program that provides the Frobenius-norm initialization in the matrix case (Appendix H.3). H.1 Pairwise Matrix Refinement Proof sketch and order-of-magnitude calculations. The goal is to start from a Frobenius-accurate initialization ‖T^(0)−T⋆‖F≤Cd3/n\| T^(0)-T \|_F≤ C d^3/n and refine it to the entrywise rate ‖T~−T⋆‖∞≤Cd/nlogC⁡d\| T-T \|_∞≤ C d/n\, ^Cd. The proof proceeds in three stages: Stage 1: Left-factor refinement via row-wise score equations. We fix the right factor A A (extracted from T^(0) T^(0) by SVD) and, for each row u∈[d]u∈[d], solve the logistic score equation Su(θ)=0S_u(θ)=0 using only comparisons involving row u. The key observation is that the score function at the truth decomposes as Su(θu⋆+δ)=Nu−Huδ−Bu+h.o.t.S_u( _u +δ)=N_u-H_uδ-B_u+h.o.t., where HuH_u is a local Hessian, NuN_u is noise, and BuB_u collects bias from the imperfect right factor and offsets. The critical order-of-magnitude calculations are: (i) each row u sees mum_u observations with cn/d≤mu≤Cn/dc\,n/d≤ m_u≤ C\,n/d; (i) the design vectors a^j a_j have norm ≤Cd−1/2≤ Cd^-1/2, so the Hessian eigenvalue satisfies λmin(Hu)≥cn/d2 _ (H_u)≥ c\,n/d^2; (i) the noise and bias are both ≤C(n/d)logC⁡d≤ C( n/d) ^Cd. The inward-pointing zero lemma (Brouwer fixed-point argument) then guarantees a solution with error ‖θ~u−θu⋆‖≤2Ruλu≤Cd2n⋅ndlogC⁡d=CdnlogC⁡d.\| θ_u- _u \|≤ 2R_u _u≤ C\, d^2n· nd ^Cd=C\, d n ^Cd. Stage 2: Recentering and curvature amplification. We project Θ~ onto the centered gauge: Θ¯=P⟂Θ~ =P_ . This preserves pairwise differences θ¯u−θ¯v=θ~u−θ~v θ_u- θ_v= θ_u- θ_v and at most doubles the row-wise error. The crucial structural advantage is that the right-factor score equation involves the pairwise Gram matrix ∑u<v(θ¯u−θ¯v)(θ¯u−θ¯v)⊤ _u<v( θ_u- θ_v)( θ_u- θ_v) , which has eigenvalue of order n (the full sample size) rather than n/d2n/d^2. This dramatic curvature amplification makes the right-factor refinement strictly easier. Stage 3: Right-factor refinement and assembly. Fixing Θ¯ , we solve column-wise score equations for each j∈[d]j∈[d]. Thanks to the amplified curvature (≥cn≥ c\,n), the right-factor error satisfies ‖A~−A⋆‖2→∞≤Cn−1/2logC⁡d\| A-A \|_2→∞≤ C\,n^-1/2 ^Cd. Combining the two factors: ‖T~−T⋆‖∞ \| T-T \|_∞ ≤‖Θ¯‖2→∞⋅‖A~−A⋆‖2→∞+‖Θ¯−Θ⋆‖2→∞⋅‖A⋆‖2→∞ ≤\| \|_2→∞·\| A-A \|_2→∞+\| - \|_2→∞·\|A \|_2→∞ ≤CdnlogC⁡d. ≤ C dn\, ^Cd. H.1.1 Problem setup, assumptions, and proof goal Constant convention. Throughout this subsection, C and c denote positive constants that may depend on the structural parameters (r,μ,κ,B0,α)(r,μ,κ,B_0,α) but not on d or n; their values may change from line to line. We consider a pairwise logistic matrix model with a rank-r signal matrix T⋆∈ℝd×dT ^d× d, factored as T⋆=Θ⋆(A⋆)⊤T = (A ) with (A⋆)⊤A⋆=Ir(A ) A =I_r, so the singular values are absorbed into Θ⋆ . The data consist of i.i.d. pairwise comparisons: each sample draws J∼Unif([d]),U,V∼Unif(u,v:1≤u<v≤d),J ([d]), \U,V\ (\\u,v\:1≤ u<v≤ d\ ), and conditional on (J,U,V)(J,U,V), Y∼Bernoulli(σ(TU,J⋆−TV,J⋆)),σ(x)=11+e−x.Y \! (σ(T _U,J-T _V,J) ), σ(x)= 11+e^-x. Since the likelihood depends only on within-column differences, we impose the gauge condition ⊤T⋆=01 T =0, which implies ⊤Θ⋆=01 =0. Our goal is to start from an initial estimator satisfying ‖T^(0)−T⋆‖F≤ΔF≤Cinitd3/n T^(0)-T _F≤ _F≤ C_init d^3/n and refine it to entrywise accuracy ‖T~−T⋆‖∞≤Cd/nlogC⁡d T-T _∞≤ C d/n\, ^Cd. Assumption H.1 (Standing assumptions). Throughout, we assume: (A1) r is fixed. (A2) Signal strength: cσd≤σr(T⋆)≤σ1(T⋆)≤Cσdc_σ\,d≤ _r(T )≤ _1(T )≤ C_σ\,d. (A3) Row-wise incoherence: ‖A⋆‖2→∞≤CAd−1/2 A _2→∞≤ C_Ad^-1/2 and ‖Θ⋆‖2→∞≤CΘd1/2 _2→∞≤ C_ d^1/2. (A4) Bounded logits: ‖T⋆‖∞≤α T _∞≤α, so 0<cα≤σ′(x)≤Cα0<c_α≤σ (x)≤ C_α and |σ′(x)|≤L3 σ (x) ≤ L_3 for all relevant predictors. (A5) The initial estimator satisfies ‖T^(0)‖∞≤C0 T^(0) _∞≤ C_0 and ‖T^(0)−T⋆‖F≤ΔF T^(0)-T _F≤ _F (after clipping if needed). (A6) Sample size: n≥CndlogC⁡dn≥ C_n\,d ^Cd for a sufficiently large constant CnC_n. H.1.2 Algorithm and proof strategy The estimation procedure is summarized in Algorithm 3. Algorithm 3 Pairwise matrix refinement 1. Data splitting. Randomly partition the data into three independent folds D1,D2,D3D_1,D_2,D_3 of equal size. 2. Initialization. Using D1D_1, compute a Frobenius-accurate initializer T^(0) T^(0) with ‖T^(0)−T⋆‖F≤ΔF\| T^(0)-T \|_F≤ _F. Extract the right factor A^∈ℝd×r A ^d× r via rank-r SVD of T^(0) T^(0), followed by row-norm projection to enforce ‖A^‖2→∞≤CAd−1/2\| A\|_2→∞≤ C_Ad^-1/2. 3. Left-factor refinement. Using D2D_2, for each row u∈[d]u∈[d], solve the row-wise logistic score equation Su(θ):=∑ℓ=1mua^jℓzℓ−σ(a^jℓ⊤θ−o^ℓ)=0S_u(θ):= _ =1^m_u a_j_ \z_ -σ( a_j_ θ- o_ ) \=0 to obtain θ~u θ_u. Recenter: Θ¯:=P⟂Θ~ :=P_ , where P⟂:=Id−d−1⊤P_ :=I_d-d^-111 . 4. Right-factor refinement. Using D3D_3, for each column j∈[d]j∈[d], solve the column-wise logistic score equation Sj(a):=∑t∈ℐjxtYt−σ(xt⊤a)=0,xt:=θ¯Ut−θ¯Vt,S_j(a):= _t _jx_t \Y_t-σ(x_t a) \=0, x_t:= θ_U_t- θ_V_t, to obtain a~j a_j. 5. Output. Return T~:=Θ¯A~⊤ T:= \, A . The proof has two symmetric halves—one for each factor update—connected by a recentering step. For the left update, we show that the row-wise score equation admits a solution near θu⋆ _u by the inward-pointing zero lemma (Lemma H.2): one establishes a deterministic sufficient condition involving local curvature, noise, and bias, then verifies these bounds uniformly over rows via concentration. The key estimates are: • Curvature: the local Hessian satisfies λmin(Hu)≥cn/d2 _ (H_u)≥ c\,n/d^2; • Noise and bias: the stochastic noise, right-factor bias, and offset bias are each ≤C((n/d)logC⁡d)≤ C (( n/d) ^Cd ); • Higher-order terms: βu _u and γu _u are small enough to close the sufficient condition. This yields ‖Θ~−Θ⋆‖2→∞≤C(d/n)logC⁡d - _2→∞≤ C(d/ n) ^Cd (Proposition H.3). After recentering (Lemma H.4), the pairwise Gram identity (Lemma H.5) ensures that the right-factor score equation has curvature of order n—far larger than the O(n/d2)O(n/d^2) curvature available for the left update. This amplification makes the right-factor refinement strictly easier, yielding ‖A~−A⋆‖2→∞≤Cn−1/2logC⁡d A-A _2→∞≤ C\,n^-1/2 ^Cd (Proposition H.6), and combining the two bounds gives the entrywise rate ‖T~−T⋆‖∞≤Cd/nlogC⁡d T-T _∞≤ C d/n\, ^Cd. H.1.3 Left-factor refinement Lemma H.2 (Inward-pointing zero lemma). Let F:ℝr→ℝrF:R^r ^r be continuous. Fix ϑ⋆∈ℝr ^r and ξ>0ξ>0. If (ϑ−ϑ⋆)⊤F(ϑ)≤0for all ‖ϑ−ϑ⋆‖=ξ,( - ) F( )≤ 0 all - =ξ, then there exists ϑ~ with F(ϑ~)=0F( )=0 and ‖ϑ~−ϑ⋆‖≤ξ - ≤ξ. Proof. If F has no zero in Bξ(ϑ⋆)B_ξ( ), define G(ϑ):=ϑ⋆+ξF(ϑ)/‖F(ϑ)‖G( ):= +ξ\,F( )/ F( ) , a continuous self-map of Bξ(ϑ⋆)B_ξ( ). By Brouwer’s theorem, G has a fixed point ϑ† on the boundary, yielding (ϑ†−ϑ⋆)⊤F(ϑ†)=ξ‖F(ϑ†)‖>0( - ) F( )=ξ F( ) >0—a contradiction. ∎ We condition on D1D_1 and work on the high-probability event ℰ1:= _1= \ ‖T^(1)−T⋆‖F≤ΔF,‖T^(1)‖∞≤C0, T^(1)-T _F≤ _F,\; T^(1) _∞≤ C_0,\; ∥A^−A⋆∥F≤CΔF/d,∥A^∥2→∞≤CAd−1/2,∥A^⊤A^−Ir∥op≤c0. A-A _F≤ C _F/d,\; A _2→∞≤ C_Ad^-1/2,\; A A-I_r _op≤ c_0 \. Let n2:=|D2|n_2:=|D_2|. Proposition H.3 (Left-factor refinement). Assume n2≥CdlogC⁡dn_2≥ C\,d ^Cd and work on ℰ1E_1. Then, with probability at least 1−d−c1-d^-c, the row-wise score equations admit solutions θ~u θ_u, u∈[d]u∈[d], with ‖Θ~−Θ⋆‖2→∞=maxu∈[d]⁡‖θ~u−θu⋆‖≤Cdn2logC⁡d. - _2→∞= _u∈[d] θ_u- _u ≤ C d n_2 ^Cd. Proof. Fix u∈[d]u∈[d]. Consider the observations in D2D_2 involving row u, reoriented so u appears on the left: triples (jℓ,wℓ,zℓ)(j_ ,w_ ,z_ ), ℓ=1,…,mu =1,…,m_u, with zℓ∣(jℓ,wℓ)∼Bernoulli(σ(ηℓ⋆)),ηℓ⋆:=(ajℓ⋆)⊤θu⋆−oℓ⋆,oℓ⋆:=Twℓjℓ⋆,o^ℓ:=T^wℓjℓ(1).z_ (j_ ,w_ ) (σ( _ )), _ :=(a_j_ ) _u -o_ , o_ :=T _w_ j_ , o_ := T^(1)_w_ j_ . Score equation and Taylor decomposition. The row-wise score equation we solve is Su(θ):=∑ℓ=1mua^jℓzℓ−σ(a^jℓ⊤θ−o^ℓ)=0.S_u(θ):= _ =1^m_u a_j_ \z_ -σ( a_j_ θ- o_ ) \=0. Since a^jℓ⊤θ−o^ℓ=ηℓ⋆+a^jℓ⊤δ+dℓ a_j_ θ- o_ = _ + a_j_ δ+d_ where δ:=θ−θu⋆δ:=θ- _u and dℓ:=(a^jℓ−ajℓ⋆)⊤θu⋆−(o^ℓ−oℓ⋆)d_ :=( a_j_ -a_j_ ) _u -( o_ -o_ ), a Taylor expansion of σ around ηℓ⋆ _ gives Su(θu⋆+δ)=Nu−Huδ−Bu(A)+Bu(O)−ℛu(δ),S_u( _u +δ)=N_u-H_uδ-B_u^(A)+B_u^(O)-R_u(δ), where Nu:=∑ℓa^jℓεℓ(εℓ:=zℓ−σ(ηℓ⋆),centered noise),Hu:=∑ℓσ′(ηℓ⋆)a^jℓa^jℓ⊤(local Hessian), N_u:= _ a_j_ _ \,( _ :=z_ -σ( _ ),\;centered noise),\,H_u:= _ σ ( _ )\, a_j_ a_j_ \,(local Hessian), Bu(A):=∑ℓσ′(ηℓ⋆)a^jℓ(a^jℓ−ajℓ⋆)⊤θu⋆(right-factor bias),Bu(O):=∑ℓσ′(ηℓ⋆)a^jℓ(o^ℓ−oℓ⋆)(offset bias), B_u^(A):= _ σ ( _ )\, a_j_ ( a_j_ -a_j_ ) _u \,(right-factor bias),\,B_u^(O):= _ σ ( _ )\, a_j_ ( o_ -o_ )\,(offset bias), and ℛu(δ):=12∑ℓa^jℓσ′(η~ℓ)(a^jℓ⊤δ+dℓ)2R_u(δ):= 12 _ a_j_ \,σ ( η_ )\,( a_j_ δ+d_ )^2 collects higher-order terms. Sufficient condition from the inward-pointing lemma. For ‖δ‖=ξ δ =ξ, the quadratic form δ⊤Su(θu⋆+δ)δ S_u( _u +δ) is bounded above by −λuξ2+L3γuξ3+Ruξ,- _uξ^2+L_3 _uξ^3+R_uξ, where λu:=λmin(Hu) _u:= _ (H_u), γu:=sup‖v‖=1∑ℓ|a^jℓ⊤v|3 _u:= _ v =1 _ a_j_ v ^3, and Ru:=‖Nu‖+‖Bu(A)‖+‖Bu(O)‖+L3βu,βu:=sup‖v‖=1∑ℓ|a^jℓ⊤v|dℓ2.R_u:= N_u + B_u^(A) + B_u^(O) +L_3 _u, _u:= _ v =1 _ a_j_ v \,d_ ^2. If Ru≤λu2/(4L3γu)R_u≤ _u^2/(4L_3 _u), the expression is non-positive at ξu=2Ru/λu _u=2R_u/ _u, and Lemma H.2 gives a zero θ~u θ_u with ‖θ~u−θu⋆‖≤2Ru/λu θ_u- _u ≤ 2R_u/ _u. Curvature. Row u appears in mu∼Bin(n2,2/d)m_u (n_2,2/d) observations. By Bernstein’s inequality applied to the scalar sum mu=∑k=1n2u∈Uk,Vkm_u= _k=1^n_21\u∈\U_k,V_k\\ (independent Bernoulli summands with mean 2/d2/d, Bernstein radius b=1b=1, and variance proxy σ2=n2⋅2/dσ^2=n_2· 2/d), a union bound over u∈[d]u∈[d] gives cn2d≤mu≤Cn2dfor all u∈[d]c\, n_2d≤ m_u≤ C\, n_2d all u∈[d] with probability at least 1−2d−101-2d^-10, provided n2≥Cdlog⁡dn_2≥ C\,d d. Since σ′(ηℓ⋆)≥cα>0σ ( _ )≥ c_α>0 and A^⊤A^⪰c0Ir A A c_0I_r on ℰ1E_1, the population Hessian satisfies [Hu∣D1]⪰cαn2d⋅1d∑j=1da^ja^j⊤⪰cn2d2Ir.E[H_u D_1] c_α n_2d· 1d _j=1^d a_j a_j c n_2d^2I_r. To pass from the population to the sample, we apply matrix Bernstein. The summands σ′(ηℓ⋆)a^jℓa^jℓ⊤σ ( _ )\, a_j_ a_j_ are independent (conditional on D1D_1), each with operator norm ≤Cα∥a^jℓ∥2≤CαCA2/d=:bH≤ C_α\| a_j_ \|^2≤ C_αC_A^2/d=:b_H. The matrix variance proxy is ‖∑ℓ[σ′(ηℓ⋆)2a^jℓa^jℓ⊤a^jℓa^jℓ⊤]‖op≤Cα2⋅CA2d⋅∑ℓ‖a^jℓ‖2/d≤Cn2d3. \| _ E [σ ( _ )^2\, a_j_ a_j_ a_j_ a_j_ ] \|_op≤ C_α^2· C_A^2d· _ E\| a_j_ \|^2/d≤ C\, n_2d^3. Matrix Bernstein gives ∥Hu−[Hu∣D1]∥op≤C(n2d3log⁡d+log⁡d)=o(n2d2)\|H_u-E[H_u D_1]\|_op≤ C ( n_2d^3 d+ dd )=o\! ( n_2d^2 ) for n2≥CdlogC⁡dn_2≥ C\,d ^Cd, whence λu≥cn2/d2 _u≥ c\,n_2/d^2 uniformly over u. Noise. Conditionally on D1D_1, the summands a^jℓεℓ a_j_ _ are independent with mean zero. We apply vector Bernstein’s inequality. The independent summands are Xℓ:=a^jℓεℓ∈ℝrX_ := a_j_ _ ^r, ℓ=1,…,mu =1,…,m_u. Their parameters are: • Bernstein radius: b=maxℓ⁡‖Xℓ‖≤‖a^jℓ‖⋅|εℓ|≤CAd−1/2⋅1=CAd−1/2b= _ \|X_ \|≤\| a_j_ \|·| _ |≤ C_Ad^-1/2· 1=C_Ad^-1/2. • Variance proxy: since [εℓ2]=σ(ηℓ⋆)(1−σ(ηℓ⋆))≤1/4E[ _ ^2]=σ( _ )(1-σ( _ ))≤ 1/4, σN2:=∑ℓ‖Xℓ‖2=∑ℓ‖a^jℓ‖2[εℓ2]≤14∑ℓ‖a^jℓ‖2≤Cn2d⋅CA2d=Cn2d2. _N^2:= _ E\|X_ \|^2= _ \| a_j_ \|^2E[ _ ^2]≤ 14 _ \| a_j_ \|^2≤ C\, n_2d· C_A^2d=C\, n_2d^2. Vector Bernstein states ¶(‖∑ℓXℓ‖≥t)≤(r+1)exp⁡(−t2/(2σN2+2bt/3)) (\| _ X_ \|≥ t)≤(r+1) (-t^2/(2 _N^2+2bt/3)). Setting t=C(n2/d2)log⁡d+C(d−1/2log⁡d)t=C (n_2/d^2) d+C(d^-1/2 d) and taking a union bound over u∈[d]u∈[d]: maxu∈[d]⁡‖Nu‖≤Cn2dlogC⁡d _u∈[d] N_u ≤ C n_2d\, ^Cd with probability at least 1−d−101-d^-10. Bias. The right-factor bias decomposes as Bu(A)=[Bu(A)∣D1]+(Bu(A)−[Bu(A)∣D1]).B_u^(A)=E[B_u^(A) D_1]+(B_u^(A)-E[B_u^(A) D_1]). The expectation satisfies ∥[Bu(A)∣D1]∥≤Cn2ΔA/d3/2 [B_u^(A) D_1] ≤ C\,n_2 _A/d^3/2 where ΔA:=‖A^−A⋆‖F≤CΔF/d _A:= A-A _F≤ C _F/d on ℰ1E_1. For the fluctuation, we again apply vector Bernstein. The summands are Xℓ(A):=σ′(ηℓ⋆)a^jℓ(a^jℓ−ajℓ⋆)⊤θu⋆−[Xℓ(A)∣D1]X_ ^(A):=σ ( _ ) a_j_ ( a_j_ -a_j_ ) _u -E[X_ ^(A) D_1]. These are independent, mean-zero, with: • Bernstein radius: bA≤C‖a^jℓ‖⋅‖a^jℓ−ajℓ⋆‖⋅‖θu⋆‖≤Cd−1/2⋅ΔA⋅d1/2=CΔAb_A≤ C\| a_j_ \|·\| a_j_ -a_j_ \|·\| _u \|≤ C\,d^-1/2· _A· d^1/2=C _A. • Variance proxy: σA2≤Cn2ΔA2/d2 _A^2≤ C\,n_2 _A^2/d^2. Vector Bernstein and a union bound over u give maxu⁡‖Bu(A)‖≤C(n2/d)logC⁡d _u B_u^(A) ≤ C( n_2/d) ^Cd. The offset bias is handled identically: the population term satisfies ∥[Bu(O)∣D1]∥≤Cn2ΔF/d5/2 [B_u^(O) D_1] ≤ C\,n_2 _F/d^5/2. For the fluctuation, the summands have Bernstein radius bO≤C‖a^jℓ‖⋅|o^ℓ−oℓ⋆|≤Cd−1/2⋅Ceb_O≤ C\| a_j_ \|·| o_ -o_ |≤ C\,d^-1/2· C_e where |o^ℓ−oℓ⋆|≤Ce| o_ -o_ |≤ C_e by the clipping guarantee, and variance proxy σO2≤Cn2ΔF2/d4 _O^2≤ C\,n_2 _F^2/d^4. Together, maxu⁡‖Bu(O)‖≤C(n2/d)logC⁡d _u B_u^(O) ≤ C( n_2/d) ^Cd. Higher-order terms. Since |a^j⊤v|≤Cd−1/2 a_j v ≤ Cd^-1/2 and dℓ2≤2((a^jℓ−ajℓ⋆)⊤θu⋆)2+2(o^ℓ−oℓ⋆)2d_ ^2≤ 2(( a_j_ -a_j_ ) _u )^2+2( o_ -o_ )^2, we get maxu⁡βu≤Cd−1/2logC⁡d,maxu⁡γu≤Cn2d5/2logC⁡d. _u _u≤ C\,d^-1/2 ^Cd, _u _u≤ C\, n_2d^5/2 ^Cd. Closing the sufficient condition. Collecting the bounds: Ru≤Cn2dlogC⁡d,λu2γu≥cn2d3/2log−C⁡d.R_u≤ C n_2d ^Cd, _u^2 _u≥ c\, n_2d^3/2 ^-Cd. The sufficient condition Ru≤λu2/(4L3γu)R_u≤ _u^2/(4L_3 _u) holds when n2≥CdlogC⁡dn_2≥ C\,d ^Cd. Therefore ‖θ~u−θu⋆‖≤2Ruλu≤Cd2n2⋅n2dlogC⁡d=Cdn2logC⁡d θ_u- _u ≤ 2R_u _u≤ C\, d^2n_2· n_2d ^Cd=C\, d n_2 ^Cd uniformly in u. ∎ H.1.4 Recentering and curvature amplification After the left update, we project onto the centered gauge to prepare for the right-factor update. The key observation is that pairwise differences amplify the curvature from O(n/d2)O(n/d^2) to O(n)O(n). Lemma H.4 (Projection to the centered gauge). Assume ⊤Θ⋆=01 =0 and define Θ¯=P⟂Θ~ =P_ . Then θ¯u−θ¯v=θ~u−θ~v,∀u,v∈[d], θ_u- θ_v= θ_u- θ_v,\,∀ u,v∈[d], ‖Θ¯−Θ⋆‖2→∞≤2‖Θ~−Θ⋆‖2→∞, - _2→∞≤ 2 - _2→∞, ‖Θ¯−Θ⋆‖F≤‖Θ~−Θ⋆‖F. - _F≤ - _F. Proof. Since Θ⋆=P⟂Θ⋆ =P_ , Θ¯−Θ⋆=P⟂(Θ~−Θ⋆) - =P_ ( - ). Pairwise differences are preserved because the mean cancels. For the 2→∞2→∞ bound, let E:=Θ~−Θ⋆E:= - with rows eu⊤e_u ; then (P⟂E)u⁣⋅=eu−e¯(P_ E)_u·=e_u- e where e¯=d−1∑vev e=d^-1 _ve_v, so ‖(P⟂E)u⁣⋅‖≤2maxv⁡‖ev‖ (P_ E)_u· ≤ 2 _v e_v . The Frobenius bound follows from P⟂P_ being an orthogonal projection. ∎ Lemma H.5 (Pairwise Gram identity). Let Θ=(θ1,…,θd)⊤∈ℝd×r =( _1,…, _d) ^d× r satisfy ⊤Θ=01 =0. Then ∑1≤u<v≤d(θu−θv)(θu−θv)⊤=dΘ⊤Θ. _1≤ u<v≤ d( _u- _v)( _u- _v) =d\, . Proof. The ordered sum equals 2dΘ⊤Θ−2(∑uθu)(∑uθu)⊤=2dΘ⊤Θ2d\, -2( _u _u)( _u _u) =2d\, by ⊤Θ=01 =0. Halving gives the unordered sum. ∎ The consequence is immediate: when the right-factor score equation uses pairwise differences xt=θ¯Ut−θ¯Vtx_t= θ_U_t- θ_V_t as covariates, the population covariance of xtx_t is [(θ¯U−θ¯V)(θ¯U−θ¯V)⊤]=2d−1Θ¯⊤Θ¯⪰cd⋅Ir,E\! [( θ_U- θ_V)( θ_U- θ_V) ]= 2d-1 c\,d· I_r, provided λmin(Θ¯⊤Θ¯)≥cd2 _ ( )≥ c\,d^2 (which holds when ‖Θ¯−Θ⋆‖2→∞≤cΘd - _2→∞≤ c_ d, since cσ2d2≤λmin(Θ⋆⊤Θ⋆)=σr(T⋆)2≤Cσ2d2c_σ^2d^2≤ _ ( )= _r(T )^2≤ C_σ^2d^2). Thus the curvature for the right update is of order n3⋅1≥cn_3· 1≥ c\,n—a factor of d2d^2 larger than the O(n/d2)O(n/d^2) curvature for the left update—because each pairwise difference has norm ≤Cd≤ C d rather than ≤Cd−1/2≤ C\,d^-1/2. H.1.5 Right-factor refinement Proposition H.6 (Right-factor refinement). Under the conditions of Proposition H.3 with n3≥CdlogC⁡dn_3≥ C\,d ^Cd, with probability at least 1−d−c1-d^-c, the column-wise score equations admit solutions a~j a_j, j∈[d]j∈[d], with ‖A~−A⋆‖2→∞=maxj∈[d]⁡‖a~j−aj⋆‖≤C1nlogC⁡d. A-A _2→∞= _j∈[d] a_j-a_j ≤ C 1 n ^Cd. Proof. The argument parallels the left update but is shorter because the amplified curvature makes each term easier to control. Fix j∈[d]j∈[d] and let ℐj:=t∈D3:Jt=jI_j:=\t∈ D_3:J_t=j\. By Bernstein’s inequality applied to |ℐj|=∑t∈D3Jt=j|I_j|= _t∈ D_31\J_t=j\ (independent Bernoulli summands with mean 1/d1/d), we have cn3/d≤|ℐj|≤Cn3/dc\,n_3/d≤|I_j|≤ C\,n_3/d uniformly over j with probability at least 1−2d−101-2d^-10. For t∈ℐjt _j, define xt:=θ¯Ut−θ¯Vt,xt⋆:=θUt⋆−θVt⋆,ht:=xt−xt⋆.x_t:= θ_U_t- θ_V_t, x_t := _U_t - _V_t , h_t:=x_t-x_t . Score equation. The column-wise score equation is Sj(a):=∑t∈ℐjxtYt−σ(xt⊤a)=0.S_j(a):= _t _jx_t\Y_t-σ(x_t a)\=0. Writing xt⊤a=(xt⋆)⊤aj⋆+xt⊤δ+ht⊤aj⋆x_t a=(x_t ) a_j +x_t δ+h_t a_j with δ:=a−aj⋆δ:=a-a_j , Taylor expansion gives Sj(aj⋆+δ)=Nj−Hjδ−Bj(Θ)−ℛj(δ),S_j(a_j +δ)=N_j-H_jδ-B_j^( )-R_j(δ), where Nj:=∑t∈ℐjxtεtN_j:= _t _jx_t _t is the noise, Hj:=∑t∈ℐjσ′((xt⋆)⊤aj⋆)xtxt⊤H_j:= _t _jσ ((x_t ) a_j )\,x_tx_t is the Hessian, Bj(Θ):=∑t∈ℐjσ′((xt⋆)⊤aj⋆)xt(ht⊤aj⋆)B_j^( ):= _t _jσ ((x_t ) a_j )\,x_t\,(h_t a_j ) is the left-factor bias, and ℛj(δ)R_j(δ) collects higher-order terms. As in the left update, the inward-pointing lemma gives a zero a~j a_j with ‖a~j−aj⋆‖≤2Rj/λj a_j-a_j ≤ 2R_j/ _j provided Rj≤λj2/(4L3γj)R_j≤ _j^2/(4L_3 _j). Curvature. By Lemma H.5 and the centering identity, the population Hessian satisfies [Hj∣D1,D2]⪰cαn3d⋅2d−1Θ¯⊤Θ¯⪰cn3Ir.E[H_j D_1,D_2] c_α n_3d· 2d-1\, c\,n_3\,I_r. The factor 1/d1/d from column sampling is exactly compensated by the pairwise covariance being of order d. Each summand has operator norm ≤Cα‖xt‖2≤Cd≤ C_α\|x_t\|^2≤ C\,d. The matrix variance proxy satisfies ‖∑t[xtxt⊤xtxt⊤]‖op≤Cn3d\| _tE[x_tx_t x_tx_t ]\|_op≤ C\,n_3\,d. Matrix Bernstein gives ‖Hj−[Hj]‖op≤C(n3dlog⁡d+dlog⁡d)=o(n3), H_j-E[H_j] _op≤ C ( n_3d d+d d )=o(n_3), whence λj≥cn3 _j≥ c\,n_3 uniformly over j. Noise. We apply vector Bernstein to the summands Xt:=xtεtX_t:=x_t _t. The parameters are: • Bernstein radius: b=maxt⁡‖Xt‖≤‖xt‖⋅|εt|≤Cd⋅1=Cdb= _t\|X_t\|≤\|x_t\|·| _t|≤ C d· 1=C d. • Variance proxy: σN2:=∑t‖Xt‖2=∑t‖xt‖2[εt2]≤C|ℐj|⋅d⋅14≤Cn3 _N^2:= _tE\|X_t\|^2= _t\|x_t\|^2E[ _t^2]≤ C|I_j|· d· 14≤ C\,n_3. Vector Bernstein and a union bound over j∈[d]j∈[d] yield maxj∈[d]⁡‖Nj‖≤Cn3logC⁡d _j∈[d] N_j ≤ C n_3\, ^Cd with probability at least 1−d−101-d^-10. Bias and higher-order terms. Set ϵΘ:=‖Θ¯−Θ⋆‖2→∞ _ := - _2→∞. Since ‖ht‖≤2ϵΘ h_t ≤ 2 _ and |ht⊤aj⋆|≤CϵΘ/d h_t a_j ≤ C\, _ / d, the bias satisfies maxj⁡‖Bj(Θ)‖≤C(n3ϵΘ/d)logC⁡d _j B_j^( ) ≤ C(n_3 _ /d) ^Cd. The higher-order terms satisfy maxj⁡βj≤C(n3ϵΘ2/d3/2)logC⁡d _j _j≤ C(n_3 _ ^2/d^3/2) ^Cd and maxj⁡γj≤Cn3d1/2logC⁡d _j _j≤ C\,n_3d^1/2 ^Cd. Closing the sufficient condition. Collecting the bounds: Rj≤C(n3+n3ϵΘd+n3ϵΘ2d3/2)logC⁡d,λj2γj≥cn3d1/2log−C⁡d.R_j≤ C ( n_3+ n_3 _ d+ n_3 _ ^2d^3/2 ) ^Cd, _j^2 _j≥ c\, n_3d^1/2 ^-Cd. The sufficient condition holds when n3≥CdlogC⁡dn_3≥ C\,d ^Cd and ϵΘ≤cΘd _ ≤ c_ d. Therefore ‖a~j−aj⋆‖≤2Rjλj≤C(1n3+ϵΘd+ϵΘ2d3/2)logC⁡d. a_j-a_j ≤ 2R_j _j≤ C ( 1 n_3+ _ d+ _ ^2d^3/2 ) ^Cd. By Proposition H.3, ϵΘ≤C(d/n2)logC⁡d _ ≤ C(d/ n_2) ^Cd, so ϵΘd≤Cn−1/2logC⁡dandϵΘ2d3/2≤CdnlogC⁡d, _ d≤ C\,n^-1/2 ^Cd _ ^2d^3/2≤ C\, dn ^Cd, with the latter being lower-order. This gives ‖A~−A⋆‖2→∞≤Cn−1/2logC⁡d A-A _2→∞≤ C\,n^-1/2 ^Cd. ∎ H.1.6 Final assembly Theorem H.7 (Entrywise guarantee for the refined estimator). Assume the standing assumptions in Assumption H.1, and suppose ‖T^(0)−T⋆‖F≤ΔF≤Cinitd3/n T^(0)-T _F≤ _F≤ C_init d^3/n. Let T~=Θ¯A~⊤ T= A be the three-split refinement estimator. Then, with probability at least 1−d−c1-d^-c, ‖T~−T⋆‖∞≤CdnlogC⁡d. T-T _∞≤ C dn\, ^Cd. Proof. By construction, T~−T⋆=(Θ¯−Θ⋆)(A⋆)⊤+Θ¯(A~−A⋆)⊤ T-T =( - )(A ) + ( A-A ) , so ‖T~−T⋆‖∞≤‖Θ¯−Θ⋆‖2→∞‖A⋆‖2→∞+‖A~−A⋆‖2→∞‖Θ¯‖2→∞. T-T _∞≤ - _2→∞ A _2→∞+ A-A _2→∞ _2→∞. By Proposition H.3 and Lemma H.4, ‖Θ¯−Θ⋆‖2→∞≤C(d/n)logC⁡d - _2→∞≤ C(d/ n) ^Cd. By Proposition H.6, ‖A~−A⋆‖2→∞≤Cn−1/2logC⁡d A-A _2→∞≤ C\,n^-1/2 ^Cd. Using ‖A⋆‖2→∞≤CAd−1/2 A _2→∞≤ C_A\,d^-1/2 and ‖Θ¯‖2→∞≤CΘd1/2 _2→∞≤ C_ \,d^1/2: ‖T~−T⋆‖∞≤C(dn⋅d−1/2+1n⋅d1/2)logC⁡d=CdnlogC⁡d. T-T _∞≤ C ( d n· d^-1/2+ 1 n· d^1/2 ) ^Cd=C dn\, ^Cd. ∎ H.2 Tensor Pairwise Refinement: From Frobenius Initializer to Entrywise Guarantee Proof sketch and order-of-magnitude calculations. This section extends the matrix refinement of Appendix H.1 to order-m tensors. The key new idea is to work with combined row objects xs,i⋆:=C(s)⋆⊤us,i⋆∈ℝrm−1x_s,i :=C_(s) u_s,i ^r^m-1 rather than the factor rows us,i⋆u_s,i directly. This reduces the tensor problem to a sequence of matrix-like logistic regression problems, one per mode. Constant convention. Throughout this subsection, C and c denote positive constants that may depend on the structural parameters (r,μ,κ,B0,α)(r,μ,κ,B_0,α) but not on d or n; their values may change from line to line. The proof rests on four governing scale identities (all for fixed r,mr,m): (i) Combined-row parameter scale: cd(m−1)/2≤‖xs,i⋆‖≤Cd(m−1)/2c\,d^(m-1)/2≤\|x_s,i \|≤ C\,d^(m-1)/2 (product of core scale dm/2d^m/2 and factor-row scale d−1/2d^-1/2). (i) Context-feature scale: ‖zJ−s⋆‖=∏t≠s‖ut,jt⋆‖≤Cd−(m−1)/2\|z_J_-s \|= _t≠ s\|u_t,j_t \|≤ C\,d^-(m-1)/2 (Kronecker product of m−1m-1 incoherent rows). (i) Rowwise curvature: λs≥cN⋅d−(m−1)≥cn/dm _s≥ c\,N· d^-(m-1)≥ c\,n/d^m (from the second-moment identity d−1∑zz⊤=d−(m−1)Id^-1Σ z =d^-(m-1)I, since each mode-s row sees N samples with cn/d≤N≤Cn/dc\,n/d≤ N≤ C\,n/d). (iv) Scalar nuisance RMS: ≤CdlogC⁡d/n≤ C d ^Cd/n, matching both the per-entry perturbation from imperfect features and the per-entry offset error. Stage 1: Row-wise combined-object refinement (one mode at a time). For each mode s in turn, fixing the other factors, we solve a row-wise logistic problem (H.2) for each row i∈[d]i∈[d]. The noise bound from vector Bernstein is ‖Ns,i‖≤CNlogC⁡d⋅d−(m−1)/2\|N_s,i\|≤ C N ^Cd· d^-(m-1)/2, and the bias (from imperfect features and offsets) matches this scale. The strong convexity from (i) gives a localization bound: ‖x^s,i−xs,i⋆‖≤gradient normcurvature≤CNlogC⁡d⋅d−(m−1)/2N⋅d−(m−1)=Cd(m−1)/2logC⁡dN=Cdm/2η,\| x_s,i-x_s,i \|\;≤\; gradient normcurvature\;≤\;C\, N ^Cd· d^-(m-1)/2N· d^-(m-1)\;=\;C\,d^(m-1)/2 ^CdN\;=\;C\,d^m/2η, where η:=log⁡d/nη:= d/n. Stage 2: Spectral lift from combined-row accuracy to factor accuracy. The combined-row matrix X(s)⁣⋆=Us⋆C(s)⋆⊤X^(s) =U_s C_(s) has singular values between cdm/2c\,d^m/2 and Cdm/2C\,d^m/2. The row-wise error Cdm/2ηC\,d^m/2η from Stage 1 gives Frobenius error ≤Cd⋅dm/2η≤ C d· d^m/2η. Applying Wedin’s perturbation theorem to extract the top-r left singular vectors U^s U_s and dividing by the singular value scale dm/2d^m/2, we obtain ‖U^s−Us⋆‖2→∞≤Cη.\| U_s-U_s \|_2→∞≤ C\,η. Stage 3: Core reconstruction and final entrywise bound. After refining all m modes, we reconstruct the core C by projecting ^(0) T^(0) onto the refined factors. A telescoping decomposition gives |^(J)−⋆(J)| | T(J)-T (J)| ≤‖^−⋆‖F∏s‖us,js‖⏟core error+∑s=1mCdm/2⋅η⋅d−(m−1)/2⏟factor-s error ≤ \| C-C \|_F _s\|u_s,j_s\|_core error+ _s=1^m C\,d^m/2·η· d^-(m-1)/2_factor-s error ≤Cdη. ≤ C d\,η. The final entrywise rate Cdlog⁡d/nC d d/n is independent of the tensor order m; all dimension factors cancel through the scale identities. This section extends the matrix pairwise refinement of Appendix H.1 to the general order-m tensor case. Starting from a single Frobenius-accurate tensor initializer, we refine one mode at a time using row-wise logistic estimation, and obtain entrywise accuracy ‖^−⋆‖∞≤Cdlog⁡dn. T-T _∞≤ C d dn. The final rate is independent of the tensor order m. H.2.1 Tensor model and assumptions Fix integers m≥2m≥ 2 and r≥1r≥ 1, both treated as constants. The signal tensor is ⋆=⋆×1U1⋆×2U2⋆⋯×mUm⋆∈ℝd×⋯×d,Us⋆∈ℝd×r,(Us⋆)⊤Us⋆=Ir.T =C ×_1U_1 ×_2U_2 ·s×_mU_m ^d×·s× d, U_s ^d× r, (U_s ) U_s =I_r. For each mode s, write C(s)⋆∈ℝr×rm−1C_(s) ^r× r^m-1 for the mode-s unfolding of the core ⋆C . Assumption H.8 (Tensor standing assumptions). (A1) Fixed rank/order. m and r are fixed constants. (A2) Incoherence. For each mode s, ‖Us⋆‖2,∞≤μ/d U_s _2,∞≤μ/ d for a constant μ. (A3) Signal scale. For each mode s, csigdm/2≤σr(C(s)⋆)≤‖C(s)⋆‖op≤Csigdm/2c_sig\,d^m/2≤ _r(C_(s) )≤ C_(s) _op≤ C_sig\,d^m/2. (A4) Bounded logits. ‖⋆‖∞≤B0 _∞≤ B_0. Hence there exist constants 0<cσ≤Cσ0<c_σ≤ C_σ and Lσ<∞L_σ<∞ (depending on B0B_0) such that for all t∈[−4B0,4B0]t∈[-4B_0,4B_0], cσ≤σ′(t)≤Cσc_σ≤σ (t)≤ C_σ and |σ′(t)|≤Lσ σ (t) ≤ L_σ. H.2.2 Observation model We use m+1m+1 independent data splits: D0D_0 for initialization and DsD_s for refinement of mode s, s=1,…,ms=1,…,m. Define N:=⌊n/(md)⌋N:= n/(md) . For each mode s and each active row i∈[d]i∈[d], split Ds(i)D_s(i) consists of exactly N independent comparisons (J−s,ℓ,wℓ,Yℓ)(J_-s, ,\,w_ ,\,Y_ ) for ℓ=1,…,N =1,…,N, where: (i) J−s,ℓ=(j1,ℓ,…,js−1,ℓ,js+1,ℓ,…,jm,ℓ)J_-s, =(j_1, ,…,j_s-1, ,j_s+1, ,…,j_m, ) is uniform on [d]m−1[d]^m-1; (i) wℓw_ is uniform on [d]∖i[d] \i\; (i) conditional on (J−s,ℓ,wℓ)(J_-s, ,w_ ), Yℓ∼Bernoulli(σ(j1,…,js−1,i,js+1,…,jm⋆−j1,…,js−1,wℓ,js+1,…,jm⋆)).Y_ \! (σ (T _j_1,…,j_s-1,i,j_s+1,…,j_m-T _j_1,…,j_s-1,w_ ,j_s+1,…,j_m ) ). Thus each mode-s row gets N samples with cn/d≤N≤Cn/dc\,n/d≤ N≤ C\,n/d. H.2.3 Initialization guarantee We assume the initializer built from D0D_0 satisfies ‖^(0)−⋆‖F≤A0d(m+1)/2η,η:=log⁡dn. T^(0)-T _F≤ A_0\,d^(m+1)/2\,η, η:= dn. (H.1) H.2.4 Algorithm The estimation procedure is summarized in Algorithm 4. Algorithm 4 Tensor pairwise refinement (one-sweep mode-by-mode) 1. Data splitting. Partition the data into m+1m+1 independent splits: D0D_0 for initialization and DsD_s for mode-s refinement, s=1,…,ms=1,…,m. 2. Initialization (Step 0). Using D0D_0, compute a Frobenius-accurate initializer ^(0) T^(0) satisfying (H.1). For each mode s, extract U¯s[0] U_s^[0] as the top-r left singular vectors of the mode-s unfolding T^(s)(0) T^(0)_(s), then apply the trim-and-orthonormalize map (Definition H.10) to obtain U^s[0] U_s^[0]. 3. Mode-s refinement (for s=1,…,ms=1,…,m). Given current factors U^1[s−1],…,U^m[s−1] U_1^[s-1],…, U_m^[s-1], using DsD_s, for each active row i∈[d]i∈[d]: (a) Construct the nuisance feature z^J−s(s):=⨂t≠su^t,jt[s−1]∈ℝrm−1 z^(s)_J_-s:= _t≠ s u^[s-1]_t,j_t ^r^m-1 and the clipped offset o^w,J−s(s):=Π[−2B0,2B0](T^opponent entry(0)) o^(s)_w,J_-s:= _[-2B_0,2B_0] ( T^(0)_opponent entry ). (b) Solve the constrained logistic problem x^s,i:=argminx∈ℬs∑ℓ=1N[−Yℓ((z^ℓ(s))⊤x−o^ℓ(s))+log⁡(1+e(z^ℓ(s))⊤x−o^ℓ(s))], x_s,i:= argmin_x _s _ =1^N [-Y_ (( z_ ^(s)) x- o_ ^(s) )+ (1+e^( z_ ^(s)) x- o_ ^(s) ) ], (H.2) where ℬs:=x∈ℝrm−1:‖x‖≤Mxd(m−1)/2B_s:=\x ^r^m-1: x ≤ M_xd^(m-1)/2\. Stack the row estimates into X^(s)∈ℝd×rm−1 X^(s) ^d× r^m-1, extract U¯s[s] U_s^[s] as the top-r left singular vectors of X^(s) X^(s), and apply trim-and-orthonormalize to obtain U^s[s] U_s^[s]. For t≠st≠ s, set U^t[s]:=U^t[s−1] U_t^[s]:= U_t^[s-1]. 4. Core reconstruction and output. Set U^s:=U^s[m] U_s:= U_s^[m] for all s and define ^:=^(0)×1U^1⊤⋯×mU^m⊤,^:=^×1U^1⋯×mU^m. C:= T^(0)×_1 U_1 ·s×_m U_m , T:= C×_1 U_1·s×_m U_m. H.2.5 Basic scale identities Lemma H.9 (Core / combined-row / feature scales). Under Assumptions (A2)–(A3), for every mode s and every row i: csigdm/2≤σr(C(s)⋆)≤‖C(s)⋆‖op≤Csigdm/2,‖us,i⋆‖≤μd. c_sig\,d^m/2≤ _r(C_(s) )≤ C_(s) _op≤ C_sig\,d^m/2, u_s,i ≤ μ d. (H.3) Hence the combined row object xs,i⋆:=C(s)⋆⊤us,i⋆∈ℝrm−1x_s,i :=C_(s) u_s,i ^r^m-1 satisfies ‖xs,i⋆‖≤Cd(m−1)/2 x_s,i ≤ C\,d^(m-1)/2. For any context J−sJ_-s, the feature zJ−s⋆(s):=⨂t≠sut,jt⋆∈ℝrm−1z (s)_J_-s:= _t≠ su_t,j_t ^r^m-1 satisfies ‖zJ−s⋆(s)‖≤Cd−(m−1)/2 z (s)_J_-s ≤ C\,d^-(m-1)/2. Therefore |⟨xs,i⋆,zJ−s⋆(s)⟩|≤C x_s,i ,z (s)_J_-s ≤ C. Proof. The first line is Assumptions (A3) and (A2). For the combined row, ‖xs,i⋆‖≤‖C(s)⋆‖op‖us,i⋆‖≤Csigdm/2⋅μd=Cd(m−1)/2. x_s,i ≤ C_(s) _op u_s,i ≤ C_sig\,d^m/2· μ d=C\,d^(m-1)/2. For the feature, ‖z⋆‖=∏t≠s‖ut,jt⋆‖≤(μ/d)m−1=Cd−(m−1)/2 z = _t≠ s u_t,j_t ≤(μ/ d)^m-1=C\,d^-(m-1)/2. The last bound is Cauchy–Schwarz. ∎ H.2.6 The trim-and-orthonormalize map Definition H.10 (Trim-and-orthonormalize). Given V∈ℝd×rV ^d× r, first clip row norms: clipτ(V)i,::=vi⋅min⁡1,τ/‖vi‖clip_τ(V)_i,::=v_i· \1,τ/ v_i \ with τ=2μ/dτ=2μ/ d. Set V~:=clipτ(V) V:=clip_τ(V), Q(V):=(V~⊤V~)−1/2Q(V):=( V V)^-1/2, and (V):=V~Q(V)T(V):= V\,Q(V), whenever V~⊤V~ V V is invertible. Lemma H.11 (Properties of trim-and-orthonormalize). Let U⋆∈ℝd×rU ^d× r satisfy (U⋆)⊤U⋆=Ir(U ) U =I_r and ‖U⋆‖2,∞≤μ/d U _2,∞≤μ/ d. Let V satisfy ‖V−U⋆‖F≤εF V-U _F≤ _F with εF≤c0 _F≤ c_0 for a sufficiently small constant c0c_0. (i) (V)⊤(V)=IrT(V) T(V)=I_r, ‖(V)‖2,∞≤Cμd−1/2 (V) _2,∞≤ Cμ d^-1/2, and ‖(V)−U⋆‖F≤CεF (V)-U _F≤ C _F. (i) If additionally ‖V−U⋆‖2,∞≤εrow V-U _2,∞≤ _row, then ‖(V)−U⋆‖2,∞≤C(εrow+d−1/2εF). (V)-U _2,∞≤ C ( _row+d^-1/2 _F ). (H.4) Proof. Since U⋆U has row norms ≤μ/d=τ/2≤μ/ d=τ/2, clipping is a projection onto a convex set containing U⋆U , hence non-expansive: ‖V~−U⋆‖F≤εF V-U _F≤ _F and ‖V~−U⋆‖2,∞≤εrow V-U _2,∞≤ _row (when the row-wise bound is given). Writing V~=U⋆+E V=U +E with ‖E‖F≤εF E _F≤ _F, we get ‖V~⊤V~−I‖2≤2‖E‖F+‖E‖F2≤2εF+εF2. V V-I _2≤ 2 E _F+ E _F^2≤ 2 _F+ _F^2. For c0c_0 small, this is ≤1/2≤ 1/2, so Q(V)Q(V) is well-defined with ‖Q(V)−I‖2≤CεF Q(V)-I _2≤ C _F and ‖Q(V)‖2≤2 Q(V) _2≤ 2. Part (i) follows: orthonormality is by construction; ‖(V)‖2,∞≤‖V~‖2,∞‖Q(V)‖2≤4μ/d (V) _2,∞≤ V _2,∞ Q(V) _2≤ 4μ/ d ‖(V)−U⋆‖F≤‖V~−U⋆‖F+‖V~‖2‖Q(V)−I‖2≤CεF. (V)-U _F≤ V-U _F+ V _2 Q(V)-I _2≤ C _F. Part (i) follows from ‖(V)−U⋆‖2,∞≤‖V~−U⋆‖2,∞+‖V~‖2,∞‖Q(V)−I‖2≤εrow+(2μ/d)⋅CεF. (V)-U _2,∞≤ V-U _2,∞+ V _2,∞ Q(V)-I _2≤ _row+(2μ/ d)· C _F. ∎ H.2.7 Initialization: from tensor Frobenius bound to row-bounded factors Lemma H.12 (HOSVD subspace error). For each mode s, let U¯s[0] U_s^[0] be the top-r left singular vectors of T^(s)(0) T^(0)_(s). Then there exists Rs[0]∈O(r)R_s^[0]∈ O(r) such that ‖U¯s[0]−Us⋆Rs[0]‖F≤C‖T^(s)(0)−T(s)⋆‖Fσr(C(s)⋆)≤Cdη. U_s^[0]-U_s R_s^[0] _F≤ C T_(s)^(0)-T_(s) _F _r(C_(s) )≤ C d\,η. (H.5) Proof. Standard Wedin/Davis–Kahan perturbation. Since ‖T^(s)(0)−T(s)⋆‖F=‖^(0)−⋆‖F≤A0d(m+1)/2η,σr(T(s)⋆)=σr(C(s)⋆)≥csigdm/2, T_(s)^(0)-T_(s) _F= T^(0)-T _F≤ A_0d^(m+1)/2η, _r(T_(s) )= _r(C_(s) )≥ c_sig\,d^m/2, we get C⋅d(m+1)/2η/dm/2=CdηC· d^(m+1)/2η/d^m/2=C d\,η. ∎ Gauge-reset convention. After Lemma H.12, we may perform a global gauge reset so that ‖U¯s[0]−Us⋆‖F≤Cdη U_s^[0]-U_s _F≤ C d\,η for every s. During the induction, each gauge reset is local to the active mode only, so previously established bounds remain valid. Corollary H.13 (Row-bounded initial factors). Assume n≥Cdlog⁡dn≥ Cd d so that dη≤c0 d\,η≤ c_0. Define U^s[0]:=(U¯s[0]) U_s^[0]:=T( U_s^[0]). Then (U^s[0])⊤U^s[0]=Ir,‖U^s[0]‖2,∞≤Cd−1/2,‖U^s[0]−Us⋆‖F≤Cdη.( U_s^[0]) U_s^[0]=I_r, U_s^[0] _2,∞≤ Cd^-1/2, U_s^[0]-U_s _F≤ C d\,η. Proof. Apply Lemma H.11 to V=U¯s[0]V= U_s^[0] with εF=Cdη _F=C d\,η. ∎ H.2.8 The row-wise combined-object refinement theorem This is the heart of the proof. We state and prove the result for a generic mode s. Fix s∈[m]s∈[m]. Assume we have nuisance factors U~t∈ℝd×r U_t ^d× r, t≠st≠ s, satisfying U~t⊤U~t=Ir,‖U~t‖2,∞≤CUd−1/2,‖U~t−Ut⋆‖F≤AUdη. U_t U_t=I_r, U_t _2,∞≤ C_Ud^-1/2, U_t-U_t _F≤ A_U d\,η. (H.6) Define the true combined row objects xs,i⋆:=C(s)⋆⊤us,i⋆,zJ−s⋆:=⨂t≠sut,jt⋆,z~J−s:=⨂t≠su~t,jt.x_s,i :=C_(s) u_s,i , z_J_-s := _t≠ su_t,j_t , z_J_-s:= _t≠ s u_t,j_t. Theorem H.14 (Generic mode-s row-wise refinement). Under Assumptions H.8, the initializer bound (H.1), and the nuisance-factor assumptions (H.6), there exists a constant C such that for every active row i, ‖x^s,i−xs,i⋆‖≤Cdm/2η x_s,i-x_s,i ≤ Cd^m/2η (H.7) with probability at least 1−d−121-d^-12, provided n≥Cdlog⁡dn≥ Cd d. Consequently, ‖X^(s)−X(s)⁣⋆‖2,∞≤Cdm/2η,‖X^(s)−X(s)⁣⋆‖F≤Cd(m+1)/2η. X^(s)-X^(s) _2,∞≤ Cd^m/2η, X^(s)-X^(s) _F≤ Cd^(m+1)/2η. (H.8) Proof. The proof proceeds by establishing gradient and curvature bounds for the logistic loss, then combining them through a strong-convexity localization argument. Score equation at the truth. Define the negative gradient (score) at xs,i⋆x_s,i : Ss,i(x):=−∇Ls,i(x)=∑ℓ=1Nz~ℓYℓ−σ(z~ℓ⊤x−o^ℓ).S_s,i(x):=-∇ L_s,i(x)= _ =1^N z_ \Y_ -σ( z_ x- o_ ) \. We decompose Ss,i(xs,i⋆)=Ns,i+Bs,iS_s,i(x_s,i )=N_s,i+B_s,i, where the noise Ns,i:=∑ℓz~ℓYℓ−σ(θℓ⋆)N_s,i:= _ z_ \Y_ -σ( _ )\ has independent mean-zero summands with ‖z~ℓ‖≤Cd−(m−1)/2 z_ ≤ Cd^-(m-1)/2, and the bias Bs,i:=∑ℓz~ℓσ(θℓ⋆)−σ(z~ℓ⊤xs,i⋆−o^ℓ)B_s,i:= _ z_ \σ( _ )-σ( z_ x_s,i - o_ )\ accounts for perturbation of both the design features and the offsets (with θℓ⋆:=zℓ⋆⊤xs,i⋆−oℓ⋆ _ :=z_ x_s,i -o_ ). Noise bound. We apply vector Bernstein’s inequality to the summands Xℓ:=z~ℓ(Yℓ−σ(θℓ⋆))∈ℝrm−1X_ := z_ (Y_ -σ( _ )) ^r^m-1, ℓ=1,…,N =1,…,N. These are independent and mean-zero conditional on D0D_0. Their parameters are: • Bernstein radius: b=maxℓ⁡‖Xℓ‖≤‖z~ℓ‖⋅|Yℓ−σ(θℓ⋆)|≤Cd−(m−1)/2⋅1=Cd−(m−1)/2b= _ \|X_ \|≤\| z_ \|·|Y_ -σ( _ )|≤ C\,d^-(m-1)/2· 1=C\,d^-(m-1)/2. • Variance proxy: since [(Yℓ−σ(θℓ⋆))2]=σ(θℓ⋆)(1−σ(θℓ⋆))≤1/4E[(Y_ -σ( _ ))^2]=σ( _ )(1-σ( _ ))≤ 1/4, σN2:=∑ℓ=1N‖Xℓ‖2=∑ℓ‖z~ℓ‖2[(Yℓ−σ(θℓ⋆))2]≤14∑ℓ‖z~ℓ‖2≤CNd−(m−1). _N^2:= _ =1^NE\|X_ \|^2= _ \| z_ \|^2E[(Y_ -σ( _ ))^2]≤ 14 _ \| z_ \|^2≤ C\,N\,d^-(m-1). Vector Bernstein states ¶(‖∑ℓXℓ‖≥t)≤(rm−1+1)exp⁡(−t2/(2σN2+2bt/3)) (\| _ X_ \|≥ t )≤(r^m-1+1) (-t^2/(2 _N^2+2bt/3) ). Setting t=CNd−(m−1)log⁡d+Cd−(m−1)/2log⁡dt=C N\,d^-(m-1) d+C\,d^-(m-1)/2 d and using rm−1r^m-1 is a constant: ‖Ns,i‖≤CNlogC⁡dd−(m−1)/2 N_s,i ≤ C N ^Cd\,d^-(m-1)/2 with probability at least 1−d−141-d^-14. Bias bound. The bias arises from the scalar nuisance perturbation qℓ:=xs,i⋆⊤(z~ℓ−zℓ⋆)−(o^ℓ−oℓ⋆).q_ :=x_s,i ( z_ -z_ )-( o_ -o_ ). We decompose this as qℓ=qZ,ℓ−qO,ℓq_ =q_Z, -q_O, where qZ,ℓ:=xs,i⋆⊤(z~ℓ−zℓ⋆)q_Z, :=x_s,i ( z_ -z_ ) is the design mismatch and qO,ℓ:=o^ℓ−oℓ⋆q_O, := o_ -o_ is the offset mismatch. Step 1: Lipschitz/Cauchy–Schwarz reduction. Since σ is CσC_σ-Lipschitz on the relevant bounded interval, | | σ(θℓ⋆)−σ(z~ℓ⊤xs,i⋆−o^ℓ)| σ( _ )-σ( z_ x_s,i - o_ )| =|σ(zℓ⋆⊤xs,i⋆−oℓ⋆)−σ(z~ℓ⊤xs,i⋆−o^ℓ)|≤Cσ|qℓ|. =|σ(z_ x_s,i -o_ )-σ( z_ x_s,i - o_ )|≤ C_σ|q_ |. Therefore, by Cauchy–Schwarz applied to Bs,i=∑ℓz~ℓσ(θℓ⋆)−σ(z~ℓ⊤xs,i⋆−o^ℓ)B_s,i= _ z_ \σ( _ )-σ( z_ x_s,i - o_ )\: ‖Bs,i‖≤Cσ(∑ℓ‖z~ℓ‖2)1/2(∑ℓqℓ2)1/2.\|B_s,i\|≤ C_σ ( _ \| z_ \|^2 )^1/2 ( _ q_ ^2 )^1/2. Step 2: Design mismatch population bound. Define Δt:=U~t−Ut⋆ _t:= U_t-U_t for each t≠st≠ s. By multilinearity of the Kronecker product, z~J−s−zJ−s⋆=⨂t≠su~t,jt−⨂t≠sut,jt⋆=∑q≠saq(J−s)+(higher-order cross terms), z_J_-s-z_J_-s = _t≠ s u_t,j_t- _t≠ su_t,j_t = _q≠ sa_q(J_-s)+(higher-order cross terms), where each first-order term aq(J−s)a_q(J_-s) has exactly one factor perturbed: aq(J−s):=u1,j1⋆⊗⋯⊗Δq,jq⊗⋯⊗um,jm⋆(with Δq,jq in the q-th position, q≠s).a_q(J_-s):=u_1,j_1 ·s _q,j_q ·s u_m,j_m (with $ _q,j_q$ in the $q$-th position, $q≠ s$). For the one-Δ terms, the population second moment satisfies 1dm−1∑J−s‖aq(J−s)‖2=∏t≠st≠q‖Ut⋆‖F2d⋅‖Δq‖F2d=1m−2⋅‖Δq‖F2d≤Cd−(m−2)⋅AU2dη2d=Cd−(m−2)η2, 1d^m-1 _J_-s\|a_q(J_-s)\|^2= _ subarrayct≠ s\\ t≠ q subarray \|U_t \|_F^2d· \| _q\|_F^2d=1^m-2· \| _q\|_F^2d≤ C\,d^-(m-2)· A_U^2dη^2d=C\,d^-(m-2)η^2, where we used (Ut⋆)⊤Ut⋆=Ir(U_t ) U_t =I_r and ‖Δq‖F2≤AU2dη2\| _q\|_F^2≤ A_U^2d\,η^2 from (H.6). Higher-order cross terms (with two or more Δ factors) contribute at most Cd−(m−2)η4C\,d^-(m-2)η^4, which is lower-order. The scalar design perturbation qZ(J−s):=xs,i⋆⊤(z~J−s−zJ−s⋆)q_Z(J_-s):=x_s,i ( z_J_-s-z_J_-s ) therefore satisfies 1dm−1∑J−sqZ2≤‖xs,i⋆‖2⋅1dm−1∑J−s‖z~J−s−zJ−s⋆‖2≤Cdm−1⋅d−(m−2)η2=Cdη2. 1d^m-1 _J_-sq_Z^2≤\|x_s,i \|^2· 1d^m-1 _J_-s\| z_J_-s-z_J_-s \|^2≤ C\,d^m-1· d^-(m-2)η^2=C\,d\,η^2. Step 3: Offset mismatch population bound. The offset o^ℓ=Π[−2B0,2B0](T^entry(0)) o_ = _[-2B_0,2B_0]( T^(0)_entry) is obtained by clipping the initial estimator to [−2B0,2B0][-2B_0,2B_0]. Since |oℓ⋆|=|entry⋆|≤B0≤2B0|o_ |=|T _entry|≤ B_0≤ 2B_0, the true offset lies in the clipping interval, and clipping to [−2B0,2B0][-2B_0,2B_0] is 11-Lipschitz. Therefore |qO(w,J−s)|=|o^−o⋆|≤|T^entry(0)−entry⋆|.|q_O(w,J_-s)|=| o-o |≤| T^(0)_entry-T _entry|. The conditional second moment (averaging over the opponent w and context J−sJ_-s) is 1(d−1)dm−1∑w,J−sqO2 1(d-1)d^m-1 _w,J_-sq_O^2 ≤1(d−1)dm−1∑w,J−s|T^entry(0)−entry⋆|2 ≤ 1(d-1)d^m-1 _w,J_-s| T^(0)_entry-T _entry|^2 ≤‖^(0)−⋆‖F2(d−1)dm−1≤A02dm+1η2dm=Cdη2. ≤ \| T^(0)-T \|_F^2(d-1)d^m-1≤ A_0^2\,d^m+1η^2d^m=C\,d\,η^2. Step 4: Sampled RMS concentration via scalar Bernstein. The total perturbation qℓ=qZ,ℓ−qO,ℓq_ =q_Z, -q_O, satisfies (combining Steps 2 and 3): [qℓ2]≤Cdη2.E[q_ ^2]≤ C\,d\,η^2. Moreover, |qℓ||q_ | is uniformly bounded: the design contribution satisfies |qZ,ℓ|≤‖xs,i⋆‖⋅‖z~ℓ−zℓ⋆‖≤Cd(m−1)/2⋅d−(m−1)/2=C|q_Z, |≤\|x_s,i \|·\| z_ -z_ \|≤ C\,d^(m-1)/2· d^-(m-1)/2=C, and the offset contribution satisfies |qO,ℓ|≤4B0|q_O, |≤ 4B_0. Hence |qℓ|≤C|q_ |≤ C (a constant independent of d and n). We apply scalar Bernstein’s inequality to ∑ℓ=1Nqℓ2 _ =1^Nq_ ^2. Define Wℓ:=qℓ2−[qℓ2]W_ :=q_ ^2-E[q_ ^2]. The summands WℓW_ are independent, mean-zero, with: • Bernstein radius: |Wℓ|≤|qℓ|2+[qℓ2]≤C2+Cdη2≤C|W_ |≤|q_ |^2+E[q_ ^2]≤ C^2+C\,d\,η^2≤ C (since dη2=dlog⁡d/n≤Cd\,η^2=d d/n≤ C for n≥Cdlog⁡dn≥ C\,d d). • Variance: ∑ℓ[Wℓ2]≤N⋅[qℓ4]≤N⋅C2⋅[qℓ2]≤CNdη2 _ E[W_ ^2]≤ N·E[q_ ^4]≤ N· C^2·E[q_ ^2]≤ C\,N\,d\,η^2. Scalar Bernstein gives: for t>0t>0, ¶(∑ℓqℓ2−N[qℓ2]≥t)≤exp⁡(−t2/2CNdη2+Ct/3). ( _ q_ ^2-NE[q_ ^2]≥ t )≤ \! (- t^2/2C\,N\,d\,η^2+C\,t/3 ). Setting t=C(Ndη2log⁡d+log⁡d)≤ClogC⁡dt=C( N\,d\,η^2 d+ d)≤ C ^Cd and using Ndη2=N⋅dlog⁡dn≤Cnd⋅dlog⁡dn=Clog⁡d,N\,d\,η^2=N· d dn≤ C\, nd· d dn=C d, we obtain ∑ℓqℓ2≤N[qℓ2]+ClogC⁡d≤CNdη2+ClogC⁡d≤ClogC⁡d _ q_ ^2≤ N\,E[q_ ^2]+C ^Cd≤ C\,N\,d\,η^2+C ^Cd≤ C ^Cd with probability at least 1−d−141-d^-14. Step 5: Combine. From Step 1, ‖Bs,i‖≤Cσ(∑ℓ‖z~ℓ‖2)1/2(∑ℓqℓ2)1/2\|B_s,i\|≤ C_σ( _ \| z_ \|^2)^1/2( _ q_ ^2)^1/2. Since ‖z~ℓ‖≤Cd−(m−1)/2\| z_ \|≤ C\,d^-(m-1)/2, we have ∑ℓ‖z~ℓ‖2≤CNd−(m−1) _ \| z_ \|^2≤ C\,N\,d^-(m-1). Combining with Step 4: ‖Bs,i‖≤CσCNd−(m−1)⋅ClogC⁡d=CNlogC⁡dd−(m−1)/2,\|B_s,i\|≤ C_σ C\,N\,d^-(m-1)· C ^Cd=C N ^Cd\,d^-(m-1)/2, matching the noise bound. Curvature. Since ‖U~t‖2,∞≤CUd−1/2 U_t _2,∞≤ C_Ud^-1/2, the features satisfy ‖z~J−s‖≤Cd−(m−1)/2 z_J_-s ≤ Cd^-(m-1)/2, and all predictors z~ℓ⊤x−o^ℓ z_ x- o_ lie in a bounded interval for x∈ℬsx _s, so σ′(z~ℓ⊤x−o^ℓ)≥cσ>0σ ( z_ x- o_ )≥ c_σ>0. The Hessian satisfies ∇2Ls,i(x)=∑ℓσ′(z~ℓ⊤x−o^ℓ)z~ℓz~ℓ⊤⪰cσ∑ℓz~ℓz~ℓ⊤.∇^2L_s,i(x)= _ σ ( z_ x- o_ )\, z_ z_ c_σ _ z_ z_ . The orthonormality U~t⊤U~t=Ir U_t U_t=I_r gives the population covariance 1dm−1∑J−sz~J−sz~J−s⊤=⨂t≠s(1d∑jtu~t,jtu~t,jt⊤)=d−(m−1)Irm−1. 1d^m-1 _J_-s z_J_-s z_J_-s = _t≠ s ( 1d _j_t u_t,j_t u_t,j_t )=d^-(m-1)I_r^m-1. We apply matrix Bernstein to the summands Zℓ:=z~ℓz~ℓ⊤−d−(m−1)Irm−1Z_ := z_ z_ -d^-(m-1)I_r^m-1. These are independent with mean zero. Their parameters are: • Bernstein radius: ‖Zℓ‖op≤‖z~ℓ‖2+d−(m−1)≤Cd−(m−1)\|Z_ \|_op≤\| z_ \|^2+d^-(m-1)≤ C\,d^-(m-1). • Matrix variance proxy: ‖∑ℓ[Zℓ2]‖op≤CNd−2(m−1)\| _ E[Z_ ^2]\|_op≤ C\,N\,d^-2(m-1). Matrix Bernstein (with ambient dimension rm−1r^m-1, a constant) yields ‖∑ℓz~ℓz~ℓ⊤−Nd−(m−1)I‖op≤C(Nd−2(m−1)log⁡d+d−(m−1)log⁡d)=o(Nd−(m−1)) \| _ z_ z_ -N\,d^-(m-1)I \|_op≤ C ( N\,d^-2(m-1) d+d^-(m-1) d )=o(N\,d^-(m-1)) for N≥ClogC⁡dN≥ C ^Cd, whence ∇2Ls,i(x)⪰cNd−(m−1)Irm−1for all x∈ℬs∇^2L_s,i(x) c\,N\,d^-(m-1)I_r^m-1 all x _s with probability at least 1−d−131-d^-13. Localization. By Lemma H.9, the true parameter satisfies ‖xs,i⋆‖≤Cd(m−1)/2 x_s,i ≤ C\,d^(m-1)/2, so xs,i⋆∈ℬsx_s,i _s for MxM_x large enough. The uniform strong convexity gives Ls,i(x)−Ls,i(xs,i⋆)≥−‖Ss,i(xs,i⋆)‖‖x−xs,i⋆‖+cN2dm−1‖x−xs,i⋆‖2.L_s,i(x)-L_s,i(x_s,i )≥- S_s,i(x_s,i ) x-x_s,i + c\,N2d^m-1 x-x_s,i ^2. The right-hand side is positive whenever ‖x−xs,i⋆‖ x-x_s,i exceeds 2dm−1‖Ss,i(xs,i⋆)‖/(cN)2d^m-1 S_s,i(x_s,i ) /(c\,N), so the minimizer satisfies ‖x^s,i−xs,i⋆‖≤Cdm−1N‖Ss,i(xs,i⋆)‖≤Cdm−1N⋅NlogC⁡dd−(m−1)/2=Cd(m−1)/2logC⁡dN. x_s,i-x_s,i ≤ C\,d^m-1N S_s,i(x_s,i ) ≤ C\,d^m-1N· N ^Cd\,d^-(m-1)/2=C\,d^(m-1)/2 ^CdN. Since cn/d≤N≤Cn/dc\,n/d≤ N≤ C\,n/d, this equals Cd(m−1)/2dlogC⁡dn=Cdm/2logC⁡dn=Cdm/2η⋅(log⁡d)(C−1)/2,C\,d^(m-1)/2 d ^Cdn=C\,d^m/2 ^Cdn=C\,d^m/2η·( d)^(C-1)/2, which is ≤Cdm/2η≤ C\,d^m/2η after absorbing the polylogarithmic factor into the constant C (recalling η=log⁡d/nη= d/n). This proves (H.7). The matrix bounds (H.8) follow by a union bound over rows and ∥⋅∥F≤d∥⋅∥2,∞ · _F≤ d\, · _2,∞. ∎ H.2.9 Spectral lift: from combined-row accuracy to factor accuracy Theorem H.15 (Spectral lift). Let X⋆=UM∈ℝd×pX =UM ^d× p with p=rm−1p=r^m-1, U⊤U=IrU U=I_r, ‖U‖2,∞≤μd−1/2 U _2,∞≤μ d^-1/2, and cMdm/2≤σr(M)≤‖M‖op≤CMdm/2c_Md^m/2≤ _r(M)≤ M _op≤ C_Md^m/2. Suppose X X satisfies ‖X^−X⋆‖2,∞≤εrow,‖X^−X⋆‖F≤εF X-X _2,∞≤ _row, X-X _F≤ _F with εF≤cdm/2 _F≤ c\,d^m/2. Let U¯ U be the top-r left singular vectors of X X. Then there exists R∈O(r)R∈ O(r) such that ‖U¯−UR‖F U-UR _F ≤CεFdm/2, ≤ C _Fd^m/2, (H.9) ‖U¯−UR‖2,∞ U-UR _2,∞ ≤C(εrowdm/2+‖X⋆‖2,∞dmεF+‖U‖2,∞dm/2εF). ≤ C ( _rowd^m/2+ X _2,∞d^m _F+ U _2,∞d^m/2 _F ). (H.10) In particular, if εrow≤Adm/2η _row≤ Ad^m/2η, εF≤Ad(m+1)/2η _F≤ Ad^(m+1)/2η, and ‖X⋆‖2,∞≤Cd(m−1)/2 X _2,∞≤ C\,d^(m-1)/2, then ‖U¯−UR‖F≤Cdη,‖U¯−UR‖2,∞≤Cη. U-UR _F≤ C d\,η, U-UR _2,∞≤ Cη. (H.11) Proof. Frobenius bound. By Wedin/Davis–Kahan: ‖sin⁡Θ(U¯,U)‖F≤CεF/σr(X⋆)≤CεF/dm/2. ( U,U) _F≤ C _F/ _r(X )≤ C _F/d^m/2. Define R:=polar(U⊤U¯)R:=polar(U U); then ‖U¯−UR‖F≤CεF/dm/2 U-UR _F≤ C _F/d^m/2, proving (H.9). Row-wise bound. Let X^r=U¯Σ^V^⊤ X_r= U V be the rank-r truncation and Er:=X^r−X⋆E_r:= X_r-X . Since X⋆X has rank r and X^r X_r is the best rank-r approximation, ‖Er‖F≤2εF E_r _F≤ 2 _F. The decomposition X^r−X⋆=(X^−X⋆)PV^+X⋆(PV^−PV) X_r-X =( X-X )P_ V+X (P_ V-P_V) with ‖PV^−PV‖2≤CεF/dm/2 P_ V-P_V _2≤ C _F/d^m/2 gives ‖Er‖2,∞≤εrow+C‖X⋆‖2,∞⋅εF/dm/2. E_r _2,∞≤ _row+C X _2,∞· _F/d^m/2. Write U¯=(X⋆+Er)V^Σ^−1=UB+ErV^Σ^−1 U=(X +E_r) V ^-1=UB+E_r V ^-1, where B:=ΣV⊤V^Σ^−1B:= V V ^-1. By Weyl, ‖Σ^−1‖2≤Cd−m/2 ^-1 _2≤ Cd^-m/2, so ‖ErV^Σ^−1‖2,∞≤Cd−m/2‖Er‖2,∞,‖U(B−R)‖2,∞≤‖U‖2,∞⋅CεF/dm/2, E_r V ^-1 _2,∞≤ Cd^-m/2 E_r _2,∞, U(B-R) _2,∞≤ U _2,∞· C _F/d^m/2, where ‖B−R‖2≤CεF/dm/2 B-R _2≤ C _F/d^m/2. Combining via U¯−UR=ErV^Σ^−1+U(B−R) U-UR=E_r V ^-1+U(B-R) gives (H.10). The concrete bounds (H.11) follow by substituting εrow=Cdm/2η _row=Cd^m/2η, εF=Cd(m+1)/2η _F=Cd^(m+1)/2η, ‖X⋆‖2,∞≤Cd(m−1)/2 X _2,∞≤ C\,d^(m-1)/2, and ‖U‖2,∞≤Cd−1/2 U _2,∞≤ C\,d^-1/2. ∎ Corollary H.16 (Factor update after spectral lift and trimming). Assume n≥Cdlog⁡dn≥ Cd d. Apply Theorem H.15 to X^(s) X^(s) and X(s)⁣⋆=Us⋆C(s)⋆X^(s) =U_s C_(s) , and let U¯s[s] U_s^[s] be the top-r left singular vectors. After a local gauge reset, define U^s[s]:=(U¯s[s]) U_s^[s]:=T( U_s^[s]). Then (U^s[s])⊤U^s[s]=Ir,‖U^s[s]‖2,∞≤Cd−1/2, ( U_s^[s]) U_s^[s]=I_r, U_s^[s] _2,∞≤ Cd^-1/2, (H.12) ‖U^s[s]−Us⋆‖F≤Cdη,‖U^s[s]−Us⋆‖2,∞≤Cη. U_s^[s]-U_s _F≤ C d\,η, U_s^[s]-U_s _2,∞≤ Cη. Proof. Theorem H.14 gives the input bounds. Theorem H.15 (with σr(C(s)⋆)≥csigdm/2 _r(C_(s) )≥ c_sig\,d^m/2 and ‖X(s)⁣⋆‖2,∞≤Cd(m−1)/2 X^(s) _2,∞≤ C\,d^(m-1)/2 from Lemma H.9) gives ‖U¯s[s]−Us⋆‖F≤Cdη U_s^[s]-U_s _F≤ C d\,η and ‖U¯s[s]−Us⋆‖2,∞≤Cη U_s^[s]-U_s _2,∞≤ Cη. Apply Lemma H.11(i) with εF=Cdη _F=C d\,η and εrow=Cη _row=Cη. ∎ H.2.10 Full-sweep entrywise guarantee Theorem H.17 (Full-sweep entrywise guarantee). Under Assumptions H.8 and the initialization guarantee (H.1), the one-sweep estimator T satisfies ‖^−⋆‖∞≤Cdlog⁡dn T-T _∞≤ C d dn (H.13) with probability at least 1−d−81-d^-8. Proof. The proof proceeds by induction over modes, followed by core reconstruction and a telescoping entrywise bound. Inductive maintenance of factor bounds. We claim that for each stage s=0,1,…,ms=0,1,…,m, after the first s modes have been refined, the current factors U^t[s] U_t^[s] satisfy (U^t[s])⊤U^t[s]=Ir,‖U^t[s]‖2,∞≤Cd−1/2,‖U^t[s]−Ut⋆‖F≤Cdη( U_t^[s]) U_t^[s]=I_r, U_t^[s] _2,∞≤ Cd^-1/2, U_t^[s]-U_t _F≤ C d\,η (H.14) for every t∈[m]t∈[m], and moreover if t≤st≤ s then the sharper row-wise bound holds: ‖U^t[s]−Ut⋆‖2,∞≤Cη. U_t^[s]-U_t _2,∞≤ Cη. (H.15) The base case s=0s=0 is Corollary H.13: the HOSVD plus trim-and-orthonormalize yields all three properties in (H.14), and (H.15) is vacuous. For the induction step s−1→s-1→ s, the nuisance factors U^t[s−1]:t≠s\ U_t^[s-1]:t≠ s\ satisfy (H.6) by the inductive hypothesis, so Theorem H.14 applies to produce the row-wise combined-object estimates X^(s) X^(s). Corollary H.16 then yields all bounds in (H.14)–(H.15) for U^s[s] U_s^[s]. For t≠st≠ s, U^t[s]=U^t[s−1] U_t^[s]= U_t^[s-1] and the gauge reset is local to mode s, so all previous bounds are preserved. Core reconstruction. Write ^−⋆=(^(0)−⋆)×1U^1⊤⋯×mU^m⊤+∑s=1mRs, C-C =( T^(0)-T )×_1 U_1 ·s×_m U_m + _s=1^mR_s, where RsR_s captures the effect of replacing Us⋆U_s by U^s U_s in the core projection. The first term has Frobenius norm ≤‖^(0)−⋆‖F≤A0d(m+1)/2η≤ T^(0)-T _F≤ A_0d^(m+1)/2η (each multiplication by U^s⊤ U_s has operator norm 11). For each RsR_s, passing to the mode-s unfolding gives ‖Rs‖F≤‖U^s⊤Us⋆−Ir‖F⋅‖C(s)⋆‖op≤Cdη⋅Csigdm/2=Cd(m+1)/2η. R_s _F≤ U_s U_s -I_r _F· C_(s) _op≤ C d\,η· C_sig\,d^m/2=Cd^(m+1)/2η. Summing over s (m is fixed): ‖^−⋆‖F≤Cd(m+1)/2η. C-C _F≤ Cd^(m+1)/2η. (H.16) Entrywise bound. Fix an entry J=(j1,…,jm)J=(j_1,…,j_m). The telescoping decomposition gives ^−⋆=(^−⋆)×1U^1⋯×mU^m+∑s=1mℰs, T-T =( C-C )×_1 U_1·s×_m U_m+ _s=1^mE_s, where ℰs:=⋆×1U^1⋯×s−1U^s−1×s(U^s−Us⋆)×s+1Us+1⋆⋯×mUm⋆E_s:=C ×_1 U_1·s×_s-1 U_s-1×_s( U_s-U_s )×_s+1U_s+1 ·s×_mU_m . The core term at entry J equals ⟨^−⋆,u^1,j1⊗⋯⊗u^m,jm⟩ C-C , u_1,j_1 ·s u_m,j_m , which is bounded by ‖^−⋆‖F∏s‖u^s,js‖≤Cd(m+1)/2η⋅d−m/2=Cdη. C-C _F _s u_s,j_s ≤ Cd^(m+1)/2η· d^-m/2=C d\,η. For each factor term ℰsE_s: |ℰs(J)|≤‖C(s)⋆‖op⋅‖u^s,js−us,js⋆‖⋅∏t<s‖u^t,jt‖⋅∏t>s‖ut,jt⋆‖. _s(J) ≤ C_(s) _op· u_s,j_s-u_s,j_s · _t<s u_t,j_t · _t>s u_t,j_t . By the inductive bounds, ‖u^s,js−us,js⋆‖≤‖U^s−Us⋆‖2,∞≤Cη u_s,j_s-u_s,j_s ≤ U_s-U_s _2,∞≤ Cη, and all other row norms are ≤Cd−1/2≤ Cd^-1/2. Hence |ℰs(J)|≤Csigdm/2⋅Cη⋅(Cd−1/2)m−1=Cdη. _s(J) ≤ C_sig\,d^m/2· Cη·(Cd^-1/2)^m-1=C d\,η. Summing over s (m is fixed): |^(J)−⋆(J)|≤Cdη=Cdlog⁡d/n T(J)-T (J) ≤ C d\,η=C d d/n. ∎ Remark H.18 (Comparison with matrix case (m=2m=2)). For m=2m=2: features zj=aj⋆∈ℝrz_j=a_j ^r with ‖zj‖≤Cd−1/2 z_j ≤ C\,d^-1/2, combined-row xi=θi⋆∈ℝrx_i= _i ^r with ‖xi‖≤Cd1/2 x_i ≤ C\,d^1/2. Core unfolding C(1)⋆∈ℝr×rC_(1) ^r× r with cd≤σr(C(1)⋆)≤Cdc\,d≤ _r(C_(1) )≤ C\,d. All of these are special cases of the general formulas with m=2m=2. The general proof reduces to Appendix H.1 when m=2m=2 and rm−1=r^m-1=r. H.3 Convex Initialization via Nuclear-Norm Penalization In the matrix case, we provide a convex initialization that achieves the Frobenius-norm rate needed by the refinement procedure. The proof follows the RSC/decomposable-regularization framework of Negahban and Wainwright (2012), adapted to the pairwise comparison setting. The three main modifications are: (i) the entry-sampling operator is replaced by the pairwise difference operator, (i) the quadratic loss is replaced by the logistic likelihood, and (i) the centered gauge ⊤M=01 M=0 removes the per-column null directions inherent to pairwise comparisons. H.3.1 Formulation Recall that each observation matrix takes the form Xt=(eut−evt)ejt⊤X_t=(e_u_t-e_v_t)e_j_t , and the negative log-likelihood of a candidate matrix M is ℒn(M):=1n∑t=1n[log⁡(1+e⟨Xt,M⟩)−yt⟨Xt,M⟩].L_n(M):= 1n _t=1^n [ (1+e X_t,M )-y_t X_t,M ]. The convex initialization procedure is summarized in Algorithm 5. Algorithm 5 Convex initialization via nuclear-norm penalized logistic regression 1. Input. Pairwise comparison data (Xt,yt)t=1n\(X_t,y_t)\_t=1^n; regularization parameter λ>0λ>0; entry bound α0≥α _0≥α; target rank r. 2. Convex program. Solve the nuclear-norm penalized logistic regression: M^∈argminM∈ℝd×dℒn(M)+λ‖M∥⋆subject to‖M‖∞≤α0,⊤M=0. M∈ argmin_M ^d× d \L_n(M)+λ M _ \ to M _∞≤ _0, 1 M=0. (H.17) 3. Rank-r projection. Compute the rank-r SVD truncation T^(0):=SVDr(M^) T^(0):=SVD_r( M). Clip entrywise: T^ij(0)←Π[−α0,α0](T^ij(0)) T^(0)_ij← _[- _0, _0]( T^(0)_ij). 4. Output. Return T^(0) T^(0) as the Frobenius-accurate initializer for Algorithm 3. Here ‖M‖⋆ M _ is the nuclear norm and α0≥α _0≥α so that the true matrix M⋆M is feasible. The entrywise constraint ensures that all logistic curvatures remain bounded below, and the centered gauge enforces identifiability. H.3.2 Population geometry of the pairwise operator The proof relies on one genuinely pairwise-specific algebraic identity. Lemma H.19 (Population pairwise quadratic form). Let X=(eu−ev)ej⊤X=(e_u-e_v)e_j with j∼Unif([d])j ([d]) and u,v\u,v\ uniform over unordered row pairs. For every Δ∈ℝd×d ^d× d with ⊤Δ=01 =0, [⟨X,Δ⟩2]=2d(d−1)‖Δ‖F2.E [ X, ^2 ]= 2d(d-1) _F^2. (H.18) Proof. Fix column j. Then ⟨X,Δ⟩=Δuj−Δvj X, = _uj- _vj. For any zero-sum vector z∈ℝdz ^d, ∑u<v(zu−zv)2=d∑u=1dzu2. _u<v(z_u-z_v)^2=d _u=1^dz_u^2. Since the jjth column of Δ has zero sum, averaging over all (d2) d2 pairs gives [(Δuj−Δvj)2∣j]=2d−1∑u=1dΔuj2.E [( _uj- _vj)^2 j ]= 2d-1 _u=1^d _uj^2. A further average over j∼Unif([d])j ([d]) yields (H.18). ∎ This is the pairwise analogue of the population norm-equivalence behind matrix completion; the centered gauge ⊤Δ=01 =0 is essential, since ⟨X,Δ⟩=0 X, =0 identically for Δ=c⊤ =1c . H.3.3 From quadratic RSC to logistic RSC The logistic loss is not quadratic, but its Bregman divergence inherits the quadratic curvature on the feasible set. Lemma H.20 (Logistic curvature reduction). Assume ‖M⋆‖∞≤α M _∞≤α and restrict to perturbations Δ with ‖M⋆+Δ‖∞≤α0 M + _∞≤ _0. Then δℒn(M⋆;Δ):=ℒn(M⋆+Δ)−ℒn(M⋆)−⟨∇ℒn(M⋆),Δ⟩≥cα02n∑t=1n⟨Xt,Δ⟩2, _n(M ; ):=L_n(M + )-L_n(M )- _n(M ), ≥ c_ _02n _t=1^n X_t, ^2, (H.19) where cα0:=inf|x|≤2α0σ′(x)>0c_ _0:= _|x|≤ 2 _0σ (x)>0. Proof. By Taylor’s theorem, for each t there exists an intermediate point ξt _t such that δℒn(M⋆;Δ)=12n∑t=1nσ′(ξt)⟨Xt,Δ⟩2. _n(M ; )= 12n _t=1^nσ ( _t) X_t, ^2. On the feasible set, |⟨Xt,M⋆⟩|≤2α| X_t,M |≤ 2α and |⟨Xt,M⋆+Δ⟩|≤2α0| X_t,M + |≤ 2 _0, so every ξt _t lies in [−2α0,2α0][-2 _0,2 _0] and σ′(ξt)≥cα0σ ( _t)≥ c_ _0. ∎ Combining Lemma H.20 with the quadratic RSC (Theorem H.21 below) immediately gives logistic RSC: on the appropriate restricted set, δℒn(M⋆;Δ)≥ccα0d2‖Δ‖F2. _n(M ; )≥ c\,c_ _0d^2 _F^2. (H.20) H.3.4 Quadratic restricted strong convexity Following Negahban and Wainwright (2012), define the restricted set of centered perturbations pw(n;c0):=Δ∈ℝd×d:⊤Δ=0,αsp(Δ)βra(Δ)≤1c0ndlog⁡d,C_pw(n;c_0):= \ ^d× d:1 =0, _sp( )\, _ra( )≤ 1c_0 nd d \, where αsp(Δ):=d‖Δ‖∞/‖Δ‖F _sp( ):=d _∞/ _F and βra(Δ):=‖Δ‖⋆/‖Δ‖F _ra( ):= _ / _F. Theorem H.21 (Pairwise quadratic RSC). There exist universal constants c0,c>0c_0,c>0 such that whenever n≥Cdlog⁡dn≥ Cd d, with probability at least 1−d−c1-d^-c, 1n∑t=1n⟨Xt,Δ⟩2≥cd2‖Δ‖F2for all Δ∈pw(n;c0). 1n _t=1^n X_t, ^2≥ cd^2 _F^2 all _pw(n;c_0). (H.21) Proof. The proof adapts the argument for Theorem 1 of Negahban and Wainwright (2012) to the pairwise difference operator. Peeling reduction. Define FΔ:=(1n∑t⟨Xt,Δ⟩2)1/2F_ :=( 1n _t X_t, ^2)^1/2 and μΔ:=2/(d(d−1))‖Δ‖F _ := 2/(d(d-1))\, _F, so that [FΔ2]=μΔ2E[F_ ^2]= _ ^2 by Lemma H.19. Since μΔ≥‖Δ‖F/d _ ≥ _F/d, it suffices to show FΔ≥μΔ/2F_ ≥ _ /2 uniformly on the restricted set. Partition the Frobenius range of the restricted set into dyadic shells SℓS_ with ‖Δ‖F∈[αℓ−1μ0,αℓμ0] _F∈[α -1 _0,\,α _0] for α=7/6α=7/6; a union bound over shells reduces the problem to a single-scale event at each level D. Discretization. At scale D, take a (D/8)(D/8)-net Δ1,…,ΔN0\ _1,…, _N_0\ of the localized set in Frobenius norm. By the Sudakov minoration inequality and nuclear-norm/operator-norm duality, the log-covering number satisfies log⁡N0≤Cρ(D)2(D/8)2d,ρ(D):=D2c0dlog⁡d/n. N_0≤ C ρ(D)^2(D/8)^2\,d, ρ(D):= D^2c_0 d d/n. The reverse triangle inequality gives FΔ≥FΔk−FΣF_ ≥ F_ _k-F_ for the nearest net element Δk _k and remainder Σ:=Δ−Δk := - _k with ‖Σ‖F≤D/8 _F≤ D/8. Net lower tail. For each fixed Δk _k, the map (⟨X1,Δk⟩,…,⟨Xn,Δk⟩)↦FΔk( X_1, _k ,…, X_n, _k ) F_ _k is (1/n)(1/ n)-Lipschitz with bounded coordinates |⟨Xt,Δk⟩|≤2/d| X_t, _k |≤ 2/d. Product-space Lipschitz concentration gives the one-sided lower tail P[FΔk<1d∥Δk∥F−t−Cdn]≤4exp⁡(−nd2t264).P [F_ _k< 1d _k _F-t- Cd n ]≤ 4 (- nd^2t^264 ). Setting t=D/(8d)t=D/(8d) and taking the union over net elements: the exponential in t absorbs log⁡N0 N_0 for c0c_0 large enough. Remainder supremum. The supremum sup‖Σ‖F≤D/8,‖Σ‖⋆≤2ρ(D)FΣ _ _F≤ D/8,\, _ ≤ 2ρ(D)F_ is controlled by symmetrization, the Ledoux–Talagrand contraction inequality (from x2x^2 to |x||x|), and operator/nuclear-norm duality, yielding [supΣFΣ2]≤CD2d2(Cradc0+116),E [ _ F_ ^2 ]≤ CD^2d^2 ( C_radc_0+ 116 ), where CradC_rad bounds the Rademacher sum [‖1n∑tεtXt‖op]≤Cradlog⁡d/(nd)E[ 1n _t _tX_t _op]≤ C_rad d/(nd) via the matrix Bernstein inequality. Choosing c0c_0 large enough makes the expectation ≤D2/(8d2)≤ D^2/(8d^2), and the product-space concentration upgrades this to a high-probability bound supΣFΣ≤D/(2d) _ F_ ≤ D/(2d). Combining the net lower tail and remainder supremum verifies the hypothesis of the peeling lemma, closing the induction over shells. ∎ H.3.5 Gradient bound and main theorem Lemma H.22 (Gradient operator-norm bound). With probability at least 1−d−c1-d^-c, ‖∇ℒn(M⋆)‖op≤Clog⁡dnd. _n(M ) _op≤ C dnd. (H.22) Proof. The gradient at the truth is ∇ℒn(M⋆)=1n∑t=1n(σ(⟨Xt,M⋆⟩)−yt)Xt. _n(M )= 1n _t=1^n (σ( X_t,M )-y_t )X_t. Each summand is zero-mean (by the model), with operator norm at most 2 2. For the right variance: [Xt⊤Xt]=(2/d)IdE[X_t X_t]=(2/d)I_d, and [(σ(⟨Xt,M⋆⟩)−yt)2Xt⊤Xt]⪯14[Xt⊤Xt]=12dId,E [(σ( X_t,M )-y_t)^2\,X_t X_t ] 14E[X_t X_t]= 12dI_d, since σ′(x)≤1/4σ (x)≤ 1/4 everywhere. The left variance is bounded similarly by O(1/d)O(1/d). The matrix Bernstein inequality with variance proxy O(n/d)O(n/d) and range 2 2 yields (H.22). ∎ We are now ready to prove the main initialization theorem. Theorem H.23 (Convex initialization for pairwise matrix). Under the model assumptions with n≥CdlogC⁡dn≥ C\,d ^Cd and λ=Cλdlog⁡d/nλ=C_λ d d/n for an appropriate constant CλC_λ, the nuclear-norm penalized estimator (H.17) satisfies ‖M^−M⋆‖F≤Cd3logC⁡dn M-M _F≤ C d^3 ^Cdn with probability at least 1−d−c1-d^-c. Proof. Set Δ^:=M^−M⋆ := M-M and choose λ=2Clog⁡d/(nd)λ=2C d/(nd) per Lemma H.22, so that λ≥2‖∇ℒn(M⋆)‖opλ≥ 2 _n(M ) _op with high probability. Basic inequality. Since M⋆M is feasible, the optimality of M M gives δℒn(M⋆;Δ^)≤−⟨∇ℒn(M⋆),Δ^⟩+λ(‖M⋆‖⋆−‖M⋆+Δ^‖⋆). _n(M ; )≤- _n(M ), +λ ( M _ - M + _ ). By the operator/nuclear-norm duality |⟨∇ℒn(M⋆),Δ^⟩|≤λ2‖Δ^‖⋆| _n(M ), |≤ λ2 _ , and by the standard decomposability of the nuclear norm, ‖M⋆‖⋆−‖M⋆+Δ^‖⋆≤‖Δ^T‖⋆−‖Δ^T⟂‖⋆, M _ - M + _ ≤ _T _ - _T _ , where T is the tangent space of rank-r matrices at M⋆M . Combining, δℒn(M⋆;Δ^)≤3λ2‖Δ^T‖⋆−λ2‖Δ^T⟂‖⋆. _n(M ; )≤ 3λ2 _T _ - λ2 _T _ . Since δℒn≥0 _n≥ 0, this yields the cone condition ‖Δ^T⟂‖⋆≤3‖Δ^T‖⋆ _T _ ≤ 3 _T _ and, using rank(Δ^T)≤2rrank( _T)≤ 2r, ‖Δ^‖⋆≤42r‖Δ^‖F. _ ≤ 4 2r\, _F. Frobenius bound. On the event of Theorem H.21, if Δ^∈pw(n;c0) _pw(n;c_0), the logistic RSC (H.20) gives δℒn(M⋆;Δ^)≥κpw‖Δ^‖F2 _n(M ; )≥ _pw _F^2 with κpw≥ccα0/d2 _pw≥ c\,c_ _0/d^2. The basic inequality then yields κpw‖Δ^‖F2≤3λ22r‖Δ^‖F, _pw _F^2≤ 3λ2 2r\, _F, so ‖Δ^‖F≤3λ2r/(2κpw)≤Cλrd2=Cd3rlog⁡d/n _F≤ 3λ 2r/(2 _pw)≤ C\,λ r\,d^2=C d^3r d/n. If instead Δ^∉pw(n;c0) _pw(n;c_0), the cone condition and the entrywise feasibility constraint ‖Δ^‖∞≤2α0 _∞≤ 2 _0 combine to give ‖Δ^‖F≤Cα0rdlog⁡d/n _F≤ C\, _0 r\,d d/n, which is bounded by the same rate for n≥Cdlog⁡dn≥ C\,d d. ∎ Remark H.24. The rate ‖M^−M⋆‖F≤Cd3rlog⁡d/n M-M _F≤ C d^3r d/n differs from the Negahban–Wainwright matrix-completion rate Cdrlog⁡d/nC dr d/n by a factor of d. This reflects the d−2d^-2 gap between the pairwise population identity (H.18) and the matrix-completion identity [⟨X,Δ⟩2]=‖Δ‖F2E[ X, ^2]= _F^2: each pairwise comparison carries O(1/d2)O(1/d^2) of the information per entry, making the d-factor overhead unavoidable.

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