Paper deep dive

A Unified Theory of Sparse Dictionary Learning in Mechanistic Interpretability: Piecewise Biconvexity and Spurious Minima

Yiming Tang, Harshvardhan Saini, Yizhen Liao, Dianbo Liu

Year: 2024Venue: arXiv preprintArea: Mechanistic Interp.Type: TheoreticalEmbeddings: 102

Intelligence

Status: succeeded | Model: google/gemini-3.1-flash-lite-preview | Prompt: intel-v1 | Confidence: 95%

Last extracted: 3/11/2026, 12:53:44 AM

Summary

The paper introduces a unified theoretical framework for Sparse Dictionary Learning (SDL) in mechanistic interpretability, modeling it as a piecewise biconvex optimization problem. It characterizes the global solution set, identifies non-identifiability issues, and explains failure modes like feature absorption and dead neurons. The authors propose the 'Linear Representation Bench' for evaluation and 'feature anchoring' to restore identifiability and improve feature recovery.

Entities (5)

Sparse Dictionary Learning · methodology · 100%Feature Anchoring · technique · 95%Linear Representation Bench · benchmark · 95%Linear Representation Hypothesis · theoretical-framework · 95%Piecewise Biconvexity · mathematical-property · 95%

Relation Signals (3)

Linear Representation Bench → evaluates → Sparse Dictionary Learning

confidence 95% · We design the Linear Representation Bench, a synthetic benchmark... to evaluate SDL methods

Sparse Dictionary Learning → exhibits → Piecewise Biconvexity

confidence 95% · We develop the first unified theoretical framework that casts all major SDL variants as a single piecewise biconvex optimization problem

Feature Anchoring → improves → Sparse Dictionary Learning

confidence 95% · we propose feature anchoring, a novel technique that restores SDL identifiability, substantially improving feature recovery

Cypher Suggestions (0)

No Cypher suggestions yet.

Abstract

Abstract:As AI models achieve remarkable capabilities across diverse domains, understanding what representations they learn and how they encode concepts has become increasingly important for both scientific progress and trustworthy deployment. Recent works in mechanistic interpretability have widely reported that neural networks represent meaningful concepts as linear directions in their representation spaces and often encode diverse concepts in superposition. Various sparse dictionary learning (SDL) methods, including sparse autoencoders, transcoders, and crosscoders, are utilized to address this by training auxiliary models with sparsity constraints to disentangle these superposed concepts into monosemantic features. These methods are the backbone of modern mechanistic interpretability, yet in practice they consistently produce polysemantic features, feature absorption, and dead neurons, with very limited theoretical understanding of why these phenomena occur. Existing theoretical work is limited to tied-weight sparse autoencoders, leaving the broader family of SDL methods without formal grounding. We develop the first unified theoretical framework that casts all major SDL variants as a single piecewise biconvex optimization problem, and characterize its global solution set, non-identifiability, and spurious optima. This analysis yields principled explanations for feature absorption and dead neurons. To expose these pathologies under full ground-truth access, we introduce the Linear Representation Bench. Guided by our theory, we propose feature anchoring, a novel technique that restores SDL identifiability, substantially improving feature recovery across synthetic benchmarks and real neural representations.

PDF

Open source PDF →Open local PDF →

Trouble viewing inline? Open PDF directly →

Full Text

102,170 characters extracted from source content.

Expand or collapse full text

A Unified Theory of Sparse Dictionary Learning in Mechanistic Interpretability: Piecewise Biconvexity and Spurious Minima Yiming Tang envelope Harshvardhan Saini Zhaoqian Yao Zheng Lin Yizhen Liao Qianxiao Li Mengnan Du Dianbo Liu envelope Abstract As AI models achieve remarkable capabilities across diverse domains, understanding what representations they learn and how they encode concepts has become increasingly important for both scientific progress and trustworthy deployment. Recent works in mechanistic interpretability have widely reported that neural networks represent meaningful concepts as linear directions in their representation spaces and often encode diverse concepts in superposition. Various sparse dictionary learning (SDL) methods, including sparse autoencoders, transcoders, and crosscoders, are utilized to address this by training auxiliary models with sparsity constraints to disentangle these superposed concepts into monosemantic features. These methods are the backbone of modern mechanistic interpretability, yet in practice they consistently produce polysemantic features, feature absorption, and dead neurons, with very limited theoretical understanding of why these phenomena occur. Existing theoretical work is limited to tied-weight sparse autoencoders, leaving the broader family of SDL methods without formal grounding. We develop the first unified theoretical framework that casts all major SDL variants as a single piecewise biconvex optimization problem, and characterize its global solution set, non-identifiability, and spurious optima. This analysis yields principled explanations for feature absorption and dead neurons. To expose these pathologies under full ground-truth access, we introduce the Linear Representation Bench. Guided by our theory, we propose feature anchoring, a novel technique that restores SDL identifiability, substantially improving feature recovery across synthetic benchmarks and real neural representations. Sparse Autoencoders, Dictionary Learning, Interpretability, Neural Networks 1 Introduction As artificial intelligence systems scale to frontier capabilities, understanding their internal mechanisms has become essential for safe deployment (Lipton, 2017; Rudin, 2019). A central insight from mechanistic interpretability is that neural networks encode interpretable concepts as linear directions in superposition (Park et al., 2024; Elhage et al., 2022), where individual neurons respond to multiple unrelated concepts—a phenomenon known as polysemanticity (Bricken et al., 2023; Templeton et al., 2024). To disentangle these superposed representations, Sparse Dictionary Learning methods, including sparse autoencoders (SAEs) (Cunningham et al., 2023), transcoders (Dunefsky et al., 2024), and crosscoders (Lindsey et al., 2024), have emerged as the dominant paradigm, achieving remarkable empirical success on frontier language models (Templeton et al., 2024; Gao et al., 2024) and enabling applications from feature steering (Wang et al., 2025) to circuit analysis (Marks et al., 2025) and medical diagnosis (Abdulaal et al., 2024). Despite this empirical success across diverse applications, SDL methods consistently exhibit persistent failure modes: learned features remain polysemantic (Chanin et al., 2025), ”dead neurons” fail to activate on any data samples (Bricken et al., 2023), and ”feature absorption” occurs where one neuron captures specific sub-concepts while another responds to the remaining related concepts (Chanin et al., 2025). Practitioners have developed techniques to address these issues, including neuron resampling (Bricken et al., 2023), auxiliary losses (Gao et al., 2024), and careful hyperparameter tuning, yet these fixes remain ad-hoc engineering solutions without principled justification. Critically, these phenomena persist even with careful training on clean data, suggesting they are not mere implementation artifacts but reflect fundamental structural properties of the SDL optimization problem itself. We argue that the root cause of these failure modes is the non-identifiability of SDL methods: even under the idealized Linear Representation Hypothesis, SDL optimization admits multiple solutions achieving perfect reconstruction loss, some recovering no interpretable ground-truth features at all, necessitating comprehensive theoretical analysis on SDL methods. While classical dictionary learning theory provides identifiability guarantees under strict conditions (Spielman et al., 2012; Gribonval and Schnass, 2010), and recent work establishes necessary conditions for tied-weight SAEs (Cui et al., 2025), no unified theoretical framework explains why diverse SDL methods, SAEs, transcoders, crosscoders, and their variants, systematically fail in predictable ways. Without theoretical grounding, the development of improved SDL methods remains largely empirical, potentially leaving highly effective techniques underexplored. In this work, we develop a unified theoretical framework that formalizes SDL as a general optimization problem, encompassing various SDL methods (Bussmann et al., 2024, 2025; Tang et al., 2025b) as special cases. We demonstrate how these diverse methods instantiate our framework through different choices of input-output representation pairs, activation functions, and loss designs. We establish rigorous conditions under which SDL methods provably recover ground-truth interpretable features, characterizing the roles of feature sparsity, latent dimensionality, and activation functions. Through detailed analysis of the optimization landscape, we demonstrate that global minima correspond to correct feature recovery and provide necessary and sufficient conditions for achieving zero loss. We establish the prevalence of spurious partial minima exhibiting un-disentangled polysemanticity, providing novel theoretical explanations for feature absorption (Chanin et al., 2025) and the effectiveness of neuron resampling (Bricken et al., 2023). We design the Linear Representation Bench, a synthetic benchmark that strictly follows the Linear Representation Hypothesis, to evaluate SDL methods with fully accessible ground-truth features. Motivated by our theoretical insights, we propose feature anchoring, a technique applicable to all SDL methods which achieves improved feature recovery by constraining learned features to known anchor directions. Our main contributions are as follows: • We build the first theoretical framework for SDL in mechanistic interpretability as a general optimization problem encompassing diverse SDL methods. • We theoretically prove that SDL optimization is biconvex, bridging mechanistic interpretability methods with traditional biconvex optimization theory. • We characterize SDL optimization landscape and prove its non-identifiability, providing novel explanations for various phenomena observed empirically. • We design the Linear Representation Bench, a benchmark with fully accessible ground-truth features, enabling fully transparent evaluation of SDL methods. • We propose a novel technique, feature anchoring, that can achieve largely improved feature recovery performance applicable for all SDL methods. We validate the effectiveness of feature anchoring with extensive experiments across diverse SDL methods and settings. 2 Preliminaries In this section, we present a unified theoretical framework for Sparse Dictionary Learning (SDL). We begin with the formal definitions of foundational concepts and the linear representation hypothesis. Then we introduce our framework and how various SDL methods instantiate it. 2.1 Input Distribution and Model Representation Definition 2.1 (Input Distribution). Let D denote the distribution over possible inputs, X, to a neural network. For example, D could be the distribution of natural images or the distribution of text sequences (Notations in Appendix B). Definition 2.2 (Model Representation). For a given model representation x, let n∈ℕn be its dimensionality. For each s∼s , the network produces a representation vector (s)x(s), which is directly observable by running the model on s. 2.2 The Linear Representation Hypothesis Empirical studies in mechanistic interpretability have observed that neural network representations encode meaningful concepts as linear directions, often in superposition (Marks and Tegmark, 2024; Nanda et al., 2023; Jiang et al., 2024; Park et al., 2025). Following Elhage et al. (2022) and Park et al. (2024), we formalize the hypothesis as follows. Assumption 2.3 (Linear Representation Hypothesis). A model representation p∈ℝnpx_p ^n_p satisfies the Linear Representation Hypothesis (LRH) if there exists a feature function :→ℝnx:X ^n and a feature matrix Wp∈ℝnp×nW_p ^n_p× n such that: 1. Linear Decomposition: For all s∼s , p(s)=Wp(s).x_p(s)=W_px(s). 2. Non-negativity: (s)∈ℝ+nx(s) ^n_+. 3. Sparsity: There exists S∈[0,1]S∈[0,1] such that ∀i∈[n]∀ i∈[n], Prs∼⁡(xi(s)=0)≥S. _s (x_i(s)=0)≥ S. 4. Interpretability: Each component xix_i of x corresponds to a human-interpretable concept. We refer to (s)x(s) as the ground-truth features, where each xi(s)x_i(s) represents the activation level of concept i for input s. WpW_p encodes how these features are represented in px_p. Assumption 2.4 (Representation Assumptions). Following recent works (Elhage et al., 2022; Park et al., 2024), we assume the following conditions on the model representations satisfying the Linear Representation Hypothesis, which align with practical use cases: 1. Unit Norm: The feature matrix Wp∈ℝnp×nW_p ^n_p× n has unit-norm columns: ‖Wp[:,i]‖2=1∀i∈[n]\|W_p[:,i]\|_2=1 ∀ i∈[n] 2. Extreme Sparsity: The sparsity level S approaches 1: S→1S→ 1 3. Independence: The ground-truth features satisfy: For any input s∼s , the features xi(s)i=1n\x_i(s)\_i=1^n are mutually independent and independent of the sparsity level S. 4. Bounded Interference: Define the maximum interference as: M:=maxi≠j⁡⟨Wp[:,i],Wp[:,j]⟩M:= _i≠ j W_p[:,i],W_p[:,j] where Wp[:,i]W_p[:,i] denotes the i-th column of WpW_p. In typical superposition scenarios, M>0M>0 but remains small (close to zero), characterizing a regime where many interpretable features are compressed into fewer dimensions with bounded mutual interference. 2.3 General Optimization Framework for SDL Now we formalize SDL as an optimization problem under the Linear Representation Hypothesis (Assumption 2.3) and the Representation Assumptions (Assumption 2.4). Definition 2.5 (Sparse Dictionary Learning). A SDL model maps an input representation p(s)∈ℝnpx_p(s) ^n_p to a target representation r(s)∈ℝnrx_r(s) ^n_r through a two-layer architecture: (i) An encoder layer that maps p(s)x_p(s) to a latent space: q(s)=σ(WEp(s))x_q(s)=σ(W_Ex_p(s)) (1) (i) A decoder layer that maps the latents to r(s)x_r(s): ^r(s)=WDq(s) x_r(s)=W_Dx_q(s) (2) where q(s)∈ℝnqx_q(s) ^n_q, WE∈ℝnq×npW_E ^n_q× n_p, WD∈ℝnr×nqW_D ^n_r× n_q, and σ:ℝnq→ℝnqσ:R^n_q ^n_q is a sparsity-inducing activation function. The SDL objective minimizes mean square error: ℒSDL=s∼[‖r(s)−WDσ(WEp(s))‖22]L_SDL=E_s [\|x_r(s)-W_Dσ(W_Ex_p(s))\|_2^2 ] (3) Under the Linear Representation Hypothesis (Assumption 2.3), both representations admit linear decompositions p(s)=Wp(s)x_p(s)=W_px(s) and r(s)=Wr(s)x_r(s)=W_rx(s) in terms of ground-truth features (s)x(s). The loss can thus be expressed as: ℒSDL=s∼[‖Wr(s)−WDσ(WEWp(s))‖22]L_SDL=E_s [\|W_rx(s)-W_Dσ(W_EW_px(s))\|_2^2 ] (4) 2.4 Instantiations: Existing SDL Methods We now demonstrate how existing SDL methods instantiate our framework through adopting different choices of input-target pairs (p,r)(x_p,x_r) and activation functions σ, and proposing variants on the loss function ℒSDLL_SDL (See Appendix A). Sparse Autoencoders (SAEs). SAEs (Cunningham et al., 2023) decompose polysemantic activations into monosemantic components through sparsity constraints. In our framework, SAEs are characterized by setting r=px_r=x_p (self-reconstruction). The encoder projects to a higher-dimensional sparse latent space, encouraging q(s)x_q(s) to capture the underlying ground-truth features (s)x(s) (Figure 1). Figure 1: Sparse Autoencoder: encoder WEW_E maps px_p to sparse latents qx_q, decoder WDW_D reconstructs from qx_q. Transcoders. Transcoders (Dunefsky et al., 2024; Paulo et al., 2025) capture interpretable features in layer-to-layer transformations. Unlike SAEs, transcoders approximate the input-output function of a target component, such as a MLP, using a sparse bottleneck. In our proposed theoretical framework, transcoders set p=mid(s)x_p=x_mid(s) and r=pre(s)x_r=x_pre(s), where mid(s)x_mid(s) denotes the inputs of one MLP block, and pre(s)x_pre(s) denotes the prediction of MLP’s outputs (Figure 2). Figure 2: Transcoder: encoder WEW_E maps p(s)x_p(s) to sparse latents q(s)x_q(s), decoder WDW_D gives r(s)x_r(s) as a prediction of MLP’s output. Crosscoders. Crosscoders (Lindsey et al., 2024) discover shared features across multiple representation sources by jointly encoding and reconstructing concatenated representations. In our framework, crosscoders set p=[p(1);…;p(m)]x_p=[x_p^(1);…;x_p^(m)] and r=[r(1);…;r(m)]x_r=[x_r^(1);…;x_r^(m)] where each superscript denotes a different source (Figure 3). Figure 3: Crosscoder: encoder WEW_E maps concatenated multi-layer input px_p to qx_q, decoder WDW_D reconstructs multi-layer output rx_r. Variants of SDL Methods. Various SDL methods fit into our theoretical framework but differ in their choices of activation functions and loss designs. Bricken et al. (2023) and Templeton et al. (2024) use ReLU activation σReLU(z)=max⁡(0,z) _ReLU(z)= (0,z) with L1L_1 regularization on latents: ℒ=s∼[‖r(s)−WDσReLU(WEp(s))‖22+λ‖q(s)‖1]L=E_s [\|x_r(s)-W_D _ReLU(W_Ex_p(s))\|_2^2+λ\|x_q(s)\|_1 ] (5) Rajamanoharan et al. (2024b) propose to utilize JumpReLU as the activation function: σJumpReLU(z)=z⋅H(z−θ) _JumpReLU(z)=z· H(z-θ) (6) combined with a smoothed Heaviside function to directly penalize the L0L_0 norm. Makhzani and Frey (2014) introduce Top-k activation: σTop-k(z)i=ziif zi is among the k largest components0otherwise _Top-k(z)_i= casesz_i&if z_i is among the k largest components\\ 0&otherwise cases (7) Bussmann et al. (2024) extend this to Batch Top-k activation, which applies Top-k selection across a batch to allow different samples to have different numbers of activated features. Rajamanoharan et al. (2024a) use ReLU activation with an additional Heaviside gating function. Gao et al. (2024) employ Top-k activation and introduce an auxiliary loss using Top-kauxk_aux dead latents: ℒ=s∼[∥r(s)−WDσTop-k(WEp(s))∥22 =E_s [\|x_r(s)-W_D _Top-k(W_Ex_p(s))\|_2^2 (8) +λaux∥p(s)−WD′σTop-kaux(WEp(s))∥22] + _aux\|x_p(s)-W_D _Top-k_aux(W_Ex_p(s))\|_2^2 ] to resurrect dead neurons. Bussmann et al. (2025) and Tang et al. (2025b) use multiple k values with a multi-scale loss: ℒ=∑i=1mλis∼[‖r(s)−WDσTop-ki(WEp(s))‖22]L= _i=1^m _iE_s [\|x_r(s)-W_D _Top-k_i(W_Ex_p(s))\|_2^2 ] (9) that sums reconstruction errors of different sparsity levels. The variants described above demonstrate that diverse SDL methods can be unified under our general framework (Definition 2.5) through specific choices of activation functions σ and loss modifications. Critically, this unification enables subsequent theoretical analysis to apply to the entire family of SDL methods, rather than just a single architecture. We also state a key property for SDL activation functions: σ(z)i∈0,zi∀i∈[nq]σ(z)_i∈\0,z_i\ ∀ i∈[n_q] (10) This property holds for ReLU, JumpReLU, Top-k, Batch Top-k, and their compositions, the primary sparsity mechanisms in practice. Notably, all these activation functions can be expressed in the form σJump(z;c) _Jump(z;c) for some threshold c≥0c≥ 0: σJump(z;c)i=zi⋅(zi>c) _Jump(z;c)_i=z_i·1(z_i>c). 3 Theoretical Results Despite the empirical success of SDL methods across diverse applications, a significant gap exists between their practical use and our theoretical understanding of their optimization dynamics. This gap has important consequences: practitioners employ techniques like dead neuron resampling (Bricken et al., 2023) and observe phenomena like feature absorption (Chanin et al., 2025) without rigorous explanations for why these occur or how to systematically address them. Without theoretical grounding, the development of improved SDL methods remains largely empirical, potentially leaving highly effective techniques underexplored. Our theoretical analysis bridges this gap by characterizing the SDL optimization landscape under the Linear Representation Hypothesis. Section 3.1 establishes a loss approximation enabling tractable analysis (Theorem 3.1). Section 3.2 proves SDL exhibits piecewise biconvex structure within activation pattern regions (Theorem 3.2). Section 3.3 characterizes the global minimum and shows the optimization is underdetermined (Theorem 3.3 and 3.4). Section 3.4 establishes the prevalence of spurious partial minima exhibiting polysemanticity (Theorem 3.7). Section 3.5 shows that hierarchical concept structures naturally induce feature absorption patterns that manifest as partial minima (Theorem 3.10). We provide full proofs in Appendix I. 3.1 Approximate Loss with Feature Reconstruction Under extreme sparsity, the SDL loss decomposes into independent per-feature reconstruction terms. Theorem 3.1 (Loss Approximation). Under Assumptions 2.3 and 2.4, define the approximate loss as: ℒ~SDL(WD,WE):=∑d=1nMd‖wrd−WDσ(WEwpd)‖2 L_SDL(W_D,W_E):= _d=1^nM_d \|w_r^d-W_Dσ(W_Ew_p^d) \|^2 (11) where Md=Pr⁡((s)=xd(s)ed)⋅[xd(s)2|xd(s)>0]M_d= (x(s)=x_d(s)e_d)·E[x_d(s)^2|x_d(s)>0]. Then |ℒSDL−ℒ~SDL|≤O((1−S)2)|L_SDL- L_SDL|≤ O((1-S)^2) (12) This approximation isolates the contribution of each ground-truth feature, making the optimization landscape amenable to convex analysis techniques applied subsequently. 3.2 SDL Optimization is Piecewise Biconvex While the SDL loss is non-convex globally due to activation discontinuities, it exhibits favorable convex structure within regions of fixed activation patterns. Theorem 3.2 (Bi-convex Structure of SDL). Consider the approximate SDL loss ℒ~SDL(WD,WE) L_SDL(W_D,W_E). Define the activation pattern region as Ω=WE∈ℝnq×np:∀d∈[n],(d)=i∈[nq]:⟨wEi,wpd⟩>c _A=\W_E ^n_q× n_p:∀ d∈[n],A(d)=\i∈[n_q]: w_E^i,w_p^d >c\\ where c≥0c≥ 0 is the activation threshold and (d)⊆[nq]A(d) [n_q] denotes neurons activated by feature d. Then ℒ~SDL L_SDL exhibits bi-convex structure over ℝnr×nq×ΩR^n_r× n_q× _A: 1. For any fixed WE∈ΩW_E∈ _A, WD↦ℒ~SDL(WD,WE)W_D L_SDL(W_D,W_E) is convex in WDW_D. 2. For any fixed WD∈Rnr×nq,WE↦ℒ~SDL(WD,WE)W_D∈ R^nr nq,W_E L_SDL(W_D,W_E) is convex in WEW_E over Ω _A. This establishes SDL as a biconvex optimization problem, bridging mechanistic interpretability with classical biconvex optimization theory. This characterization enables both theoretical analysis of the optimization landscape and implementation of specialized biconvex methods for SDL. 3.3 Characterizing the Global Minimum of SDL Characterizing the global minimum reveals both the success conditions for SDL training and the fundamental underdetermined nature of the optimization problem. Theorem 3.3 (Successful Reconstruction Achieves Near-Zero Loss). Consider the approximate SDL loss ℒ~SDL(WD,WE)=∑d=1nMd‖wrd−WDσ(WEwpd)‖2 L_SDL(W_D,W_E)= _d=1^nM_d \|w_r^d-W_Dσ(W_Ew_p^d) \|^2 (13) where Md=Pr⁡((s)=xd(s)ed)⋅[xd(s)2|xd(s)>0]M_d= (x(s)=x_d(s)e_d)·E[x_d(s)^2|x_d(s)>0]. When nq≥n_q≥ n and σ satisfies σ(z)i∈0,ziσ(z)_i∈\0,z_i\ for all i, the configuration WD∗=[Wr,],WE∗=[Wp⊤]W_D^*=[W_r,0], W_E^*= bmatrixW_p \\ 0 bmatrix (14) where 0 denotes zero padding to dimension nqn_q, satisfies: ℒ~SDL(WD∗,WE∗)=O((1−S)M2) L_SDL(W_D^*,W_E^*)=O((1-S)M^2) (15) where M is the maximum interference. Theorem 3.4 (Necessary and Sufficient Conditions for Zero Loss). The approximate loss satisfies ℒ~SDL(WD,WE)=0 L_SDL(W_D,W_E)=0 if and only if wrd=WDσ(WEwpd)for all d∈[n]w_r^d=W_Dσ(W_Ew_p^d) all d∈[n] (16) Theorem 3.3 provides a constructive global minimum that recovers ground-truth features, while Theorem 3.4 reveals the complete solution space. This system of n vector equations is underdetermined when nq>n_q>n, admitting multiple solutions beyond the feature-recovering configuration. Critically, some solutions achieve zero reconstruction loss without recovering any ground-truth features (Figure 4). Figure 4: Zero reconstruction loss without recovering ground-truth features. We design the Linear Representation Bench that enable full knowledge of the ground truth features to study SDL methods. We observe one concerning phenomenon that these methods can achieve zero loss without recovering any ground truth features. Left: Four ground-truth feature directions. Middle: Learned encoder directions fail to align with ground truth. Right: Learned decoder directions are rotated accordingly. Although ℒ~SDL≈0 L_SDL≈ 0, the learned features bear no correspondence to interpretable ground-truth concepts, demonstrating the underdetermined nature of SDL optimization. 3.4 Characterizing Spurious Partial Minima of SDL Beyond the global minimum, SDL optimization exhibits spurious partial minima where neurons exhibit polysemanticity—responding to multiple unrelated features. Example 3.5 (Spurious Partial Minimum). Consider n=np=nq=nr=2n=n_p=n_q=n_r=2, σ=σTop-1⋅σReLUσ= _Top-1· _ReLU, S→1S→ 1, and M1=M2=1M_1=M_2=1. Let: wp1=[10],wp2=[01],wr1=[10],wr2=[01]w_p^1= bmatrix1\\ 0 bmatrix, w_p^2= bmatrix0\\ 1 bmatrix, w_r^1= bmatrix1\\ 0 bmatrix, w_r^2= bmatrix0\\ 1 bmatrix (17) Consider the configuration where neuron 1 activates for both features while neuron 2 remains dead: WE∗=[1100],WD∗=[1/201/20]W_E^*= bmatrix1&1\\ 0&0 bmatrix, W_D^*= bmatrix1/2&0\\ 1/2&0 bmatrix (18) Within the activation pattern region Ω=WE:⟨wE1,wp1⟩>0,⟨wE1,wp2⟩>0 _A=\W_E: w_E^1,w_p^1 >0, w_E^1,w_p^2 >0\, direct calculation shows ∇WDℒ~=0 _W_D L=0 and ∇WEℒ~=0 _W_E L=0. By traditional biconvex optimization theory (Gorski et al., 2007), (WD∗,WE∗)(W_D^*,W_E^*) is a partial optimum. However, this configuration exhibits polysemanticity with suboptimal loss: ℒ~SDL(WD∗,WE∗)=1>0 L_SDL(W_D^*,W_E^*)=1>0. Definition 3.6 (Activation Pattern). An activation pattern is a collection =(ℱ1,…,ℱN)P=(F_1,…,F_N) where ℱi⊆[n]F_i [n] denotes the set of ground-truth features that activate neuron i. An activation pattern P is called: • Polysemantic if ∃i∈[N]∃ i∈[N] such that |ℱi|≥2|F_i|≥ 2 (at least one neuron responds to multiple features). • Realizable if there exists an encoder σ(WE(⋅))∈ℝN×npσ(W_E(·)) ^N× n_p such that: ∀i∈[N],ℱi=d∈[n]:(σ(WEwpd))i>0∀ i∈[N], _i= \d∈[n]: (σ(W_Ew_p^d) )_i>0 \ (19) That is, neuron i activates for exactly the features in ℱiF_i when they appear in isolation. Theorem 3.7 (Prevalence of Spurious Partial Minima). Under Assumptions 2.3 and 2.4 with n≥2n≥ 2 and nq≥n_q≥ n, for any activation pattern =(ℱ1,…,ℱnq)P=(F_1,…,F_n_q) that is realizable, polysemantic, and forms a partition of [n][n], there exists a partial minimum (WD∗,WE∗)(W_D^*,W_E^*) of ℒ~SDL L_SDL exhibiting this pattern with positive loss. This establishes that partial minima are pervasive in SDL optimization: every realizable polysemantic activation pattern corresponds to a partial optima point where gradient descent can become trapped, a persistent challenge for SDL. 3.5 Theoretical Explanation for Feature Absorption Feature absorption—where one neuron captures, or ”absorbs”, a specific sub-concept while another responds to remaining related concepts—frequently occurs in SDL training (Chanin et al., 2025). Though prevalently encountered and unwanted, researchers have limited understanding about why feature absorption occurs in SDL training. Here we show hierarchical concept structures naturally introduce realizable activation patterns exhibiting feature absorption and therefore connected with the framework’s partial minima. Figure 5: Feature absorption emerges from hierarchical concept structure. Left: Ideal SDL features without absorption. Right: hierarchical concept structure exists and only a proportion of the sub-concepts of ”Dog” can activate the SDL feature. Example 3.8 (Feature Absorption). Consider a representation space with a parent concept ”Dog” and four sub-concepts: ”Border Collie”, ”Golden Retriever”, ”Husky”, and ”German Shepherd”. Ideally, SDL learns separate monosemantic neurons for Dog, Cat, Horse, and Elephant, with each dog breed activating only the Dog neuron. However, feature absorption can result in the pattern in Figure 5: one neuron exclusively captures ”Border Collie” (absorbed feature), while another responds to the remaining three breeds collectively (main line interpretation). Definition 3.9 (Hierarchical Concept Structure). A set of ground-truth features exhibits hierarchical structure if it is composed of a parent concept p and a set of sub-concepts c1,…,ckc_1,…,c_k satisfying: p(x)>0⇔∃i∈[k],ci(x)>0p(x)>0 ∃ i∈[k],c_i(x)>0. Theorem 3.10 (Feature Absorption from Hierarchical Structure). Suppose there exist M parent concepts with hierarchical decompositions into sub-concepts: for each i∈[M]i∈[M], parent concept did_i decomposes into sub-concepts ℱi=di,1,…,di,kiF_i=\d_i,1,…,d_i,k_i\ where ki≥2k_i≥ 2. If the activation pattern =(ℱ1,…,ℱM)P=(F_1,…,F_M) is realizable, then ∀i∗∈[M]∀ i^*∈[M], ∃j∗∈[ki∗]∃ j^*∈[k_i^*] such that the following activation pattern exhibiting feature absorption is realizable: ′=(ℱ1,…,ℱi∗∖di∗,j∗,di∗,j∗,…,ℱM)P =(F_1,…,F_i^* \d_i^*,j^*\,\d_i^*,j^*\,…,F_M) (20) Theorem 3.10 explains why feature absorption persists even with careful training: hierarchical concept structures naturally induce realizable polysemantic patterns that manifest as partial minima, providing theoretical grounding for this widely observed empirical phenomenon. 4 Method Our theoretical analysis reveals a fundamental challenge in SDL optimization: the underdetermined nature of the loss landscape (Theorem 3.4), and the solution space admits multiple configurations—some achieving zero reconstruction loss without recovering any interpretable ground-truth features (Figure 4). These theoretical findings motivate our method, feature anchoring, a technique that constrains a subset of features to align with known semantic directions. 4.1 Anchor Feature Extraction Feature anchoring requires identifying k anchor features ~p(i),~r(i)i=1k\ w_p^(i), w_r^(i)\_i=1^k that represent semantically meaningful directions in the representation space. We present two methods for obtaining these anchors: Ground-Truth Features (Linear Representation Bench). When ground-truth features are available—as in our Linear Representation Bench where features are known by construction—we directly use a random subset of k ground-truth feature directions: w~p(i)=Wptrue[:,i],w~r(i)=Wrtrue[:,i],i∈ w_p^(i)=W_p^true[:,i], w_r^(i)=W_r^true[:,i], i (21) where ⊂[n]K⊂[n] is a randomly selected subset of size k. Subpopulation Mean Embeddings. For real-world datasets where ground-truth features are unknown, we identify semantic subpopulations Luo et al. (2024) and compute their mean representations. Specifically, given a labeled dataset =(sj,yj)j=1ND=\(s_j,y_j)\_j=1^N, where yj∈1,…,Cy_j∈\1,…,C\, we compute the mean of representations for each class: ¯p(c)=1|j:yj=c|∑j:yj=cp(sj) x_p^(c)= 1|\j:y_j=c\| _j:y_j=cx_p(s_j) (22) Then we normalize each mean representation to obtain anchor directions: ~p(c)=¯p(c)‖¯p(c)‖2 w_p^(c)= x_p^(c)\| x_p^(c)\|_2 (23) 4.2 SDL Loss Function with Feature Anchoring Given k anchor features ~p(i),~r(i)i=1k\ w_p^(i), w_r^(i)\_i=1^k, we modify the SDL optimization objective to include an anchoring penalty that constrains the first k encoder rows and decoder columns to align with these anchors. The complete anchored SDL objective is: ℒSDL-FA=ℒSDL+λanchorℒanchorL_SDL-FA=L_SDL+ _anchorL_anchor (24) where ℒSDLL_SDL is the standard SDL loss (Equation 3) and the anchoring loss is: ℒanchor=∥WE[1:k,:]−[w~p(1),…,w~p(k)]⊤∥F2+∥WD[:,1:k]−[w~r(1),…,w~r(k)]∥F2 splitL_anchor&=\|W_E[1:k,:]-[ w_p^(1),…, w_p^(k)] \|^2_F\\ & +\|W_D[:,1:k]-[ w_r^(1),…, w_r^(k)]\|^2_F split (25) This feature anchoring technique (Equation 25) reduces the underdetermined nature of SDL training, and is method-agnostic: it applies equally to SAEs, transcoders, crosscoders, and their variants (TopK SAEs, Matryoshka SAEs, etc.) since it only constrains the encoder WEW_E and decoder WDW_D matrices that are common to all SDL architectures. This universality makes feature anchoring a broadly useful technique for improving performance across SDL methods. 5 Experimental Results We first evaluate feature anchoring with various SDL methods on the Linear Representation Bench (Section 5.1), a synthetic benchmark with fully accessible ground-truth features that precisely instantiates our theoretical assumptions. Second, we validate feature anchoring on real-world data by training SDL methods on CLIP embeddings of ImageNet-1K. Third, we provide empirical evidence showing that dead neuron resampling helps escape spurious local minima in large language models. We provide comprehensive ablation studies in Appendix G. 5.1 Results on The Linear Representation Bench To validate our theoretical predictions, we design the Linear Representation Bench, a benchmark that precisely instantiates Assumptions 2.3 and 2.4 with fully known GT features. Data Generation. We generate synthetic representations p(s)=Wptrue(s)x_p(s)=W_p^truex(s) where ground-truth features (s)∈ℝnx(s) ^n follow shifted exponential distributions with sparsity S. The feature matrix Wptrue∈ℝnp×nW_p^true ^n_p× n is constructed via gradient-based optimization (Detailed in Appendix C). Metrics. Given learned features Wplearned=WE⊤∈ℝnp×nqW_p^learned=W_E ^n_p× n_q and ground truth features Wptrue∈ℝnp×nW_p^true ^n_p× n (columns are unit-norm feature directions), we compute the similarity matrix =|Wplearned⊤Wptrue|∈ℝnq×nS=|W_p^learned W_p^true| ^n_q× n. For each ground truth feature i∈[n]i∈[n], we find its best match: si=maxj⁡Sjis_i= _jS_ji. • GT Recovery is the fraction of ground truth features with si>τs_i>τ, (Wplearned)=1n∑i=1nsi>τM_GT(W_p^learned)= 1n _i=1^n1\s_i>τ\. • Maximum Inner Product is the mean of best matches: (Wplearned)=1n∑i=1nsiM_IP(W_p^learned)= 1n _i=1^ns_i. As shown in Table 1, feature anchoring significantly improves feature recovery performance for all SDL methods. Table 1: Feature recovery on the Linear Representation Bench (n=1000n=1000, np=nr=768n_p=n_r=768, nq=16000n_q=16000, =0.99S=0.99). Feature anchoring (FA) consistently improves GT Recovery and Maximum Inner Product (MIP) across all SDL methods. Method M_GT ↑ M_IP ↑ L_0-norm↓ ReLU SAE 0.00% 0.205 2787.4 + Feature Anchoring 0.00% 0.246 1715.1 JumpReLU SAE 0.00% 0.237 1572.3 + Feature Anchoring 0.00% 0.333 1139.6 TopK SAE 84.90% 0.983 54.1 + Feature Anchoring 87.63% 0.986 54.1 BatchTopK SAE 84.80% 0.981 65.0 + Feature Anchoring 89.38% 0.988 60.1 Matryoshka SAE 83.70% 0.982 58.9 + Feature Anchoring 87.32% 0.985 57.9 Transcoder 23.60% 0.838 50.6 + Feature Anchoring 25.05% 0.838 49.5 Crosscoder 56.42% 0.940 83.4 + Feature Anchoring 57.71% 0.941 116.3 5.2 Results on CLIP Embeddings of ImageNet To validate that feature anchoring generalizes beyond synthetic benchmarks to real-world representations, we apply our method to CLIP embeddings of ImageNet-1K (Russakovsky et al., 2015). We extract CLIP-ViT-B/32 (Radford et al., 2021) image embeddings for all ImageNet training images and compute ground-truth anchor features as normalized class mean embeddings: w~p(c)=x¯p(c)/‖x¯p(c)‖2 w_p^(c)= x_p^(c)/\| x_p^(c)\|_2 where x¯p(c) x_p^(c) is the mean embedding over all images in class c. We train TopK, BatchTopK, and Matryoshka SAEs with np=nr=768n_p=n_r=768, nq=16,384n_q=16,384, and use k=30k=30 anchors with λanchor=1.0 _anchor=1.0. Table 2 shows feature anchoring consistently improves feature recovery (Examples in Appendix H). Table 2: Feature recovery on the CLIP embeddings. Method M_GT ↑ M_IP ↑ L_0-norm↓ TopK SAE 0.00% 0.517 64.0 + Feature Anchoring 24.13% 0.851 64.0 BatchTopK SAE 0.00% 0.508 64.0 + Feature Anchoring 24.13% 0.847 64.0 Matryoshka SAE 0.00% 0.683 128.0 + Feature Anchoring 24.45% 0.858 128.0 5.3 Neuron Resampling Helps Escape Partial Minima Our partial minima analysis (Theorem 3.7) proves the prevalence of spurious partial minima and connects them to dead neurons (ℱi=∅F_i= ). To address this, we utilize neuron resampling (Bricken et al., 2023) to help SDL training overcome these partial minima. We argue that resampling reinitializes dead neurons toward under-reconstructed directions, perturbing the optimization away from the partial minima. To validate this, we train SAEs on Llama 3.1 8B Instruct (layer 12, dimension 4096) with latent dimension 131,072 for 30,000 steps on FineWeb-Edu (Penedo et al., 2024), processing 4096 token activations per step. We compare standard training against feature resampling after 5000 steps. As shown in Figure 6, resampling enables the optimizer to escape spurious local minima, achieving lower final loss. Figure 6: Feature resampling accelerates convergence and improves final loss. Training curves on Llama 3.1 8B comparing standard SAE training (blue) with periodic dead neuron resampling after 5000 steps (in the plot x axis is scaled by 100). 6 Conclusion We develop the first unified theoretical framework for Sparse Dictionary Learning in mechanistic interpretability, demonstrating how diverse SDL methods instantiate a single optimization problem. We prove that SDL exhibits piecewise biconvex structure, bridging mechanistic interpretability with classical optimization theory. We characterize the global minimum and establish that the optimization is fundamentally underdetermined, admitting solutions that achieve zero reconstruction loss without recovering interpretable features. We demonstrate that spurious partial minima exhibiting polysemanticity are pervasive, and prove that hierarchical concept structures naturally induce feature absorption patterns that manifest as partial minima. To validate our theory, we design the Linear Representation Bench with fully accessible ground-truth features, and propose feature anchoring—a technique applicable to all SDL methods that addresses the underdetermined nature of optimization. Impact Statement This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here. Code Availability We provide full scripts of the Linear Representation Bench and feature anchoring in this GitHub repository: The_Linear_Representation_Bench. References A. Abdulaal, H. Fry, N. Montaña-Brown, A. Ijishakin, J. Gao, S. Hyland, D. C. Alexander, and D. C. Castro (2024) An x-ray is worth 15 features: sparse autoencoders for interpretable radiology report generation. External Links: 2410.03334, Link Cited by: §D.1, §D.2, §1. M. Aharon, M. Elad, and A. Bruckstein (2006) K-svd: an algorithm for designing overcomplete dictionaries for sparse representation. IEEE Transactions on signal processing 54 (11), p. 4311–4322. Cited by: §D.2, 5th item. L. Bereska and E. Gavves (2024) Mechanistic interpretability for ai safety – a review. External Links: 2404.14082, Link Cited by: §D.1. T. Bricken, A. Templeton, J. Batson, B. Chen, A. Jermyn, T. Conerly, N. L. Turner, C. Anil, C. Denison, A. Askell, R. Lasenby, Y. Wu, S. Kravec, N. Schiefer, T. Maxwell, N. Joseph, A. Tamkin, K. Nguyen, B. McLean, J. E. Burke, T. Hume, S. Carter, T. Henighan, and C. Olah (2023) Towards monosemanticity: decomposing language models with dictionary learning. Transformer Circuits Thread. External Links: Link Cited by: §1, §1, §1, §2.4, §3, §5.3. B. Bussmann, P. Leask, and N. Nanda (2024) BatchTopK sparse autoencoders. External Links: 2412.06410, Link Cited by: §1, §2.4. B. Bussmann, N. Nabeshima, A. Karvonen, and N. Nanda (2025) Learning multi-level features with matryoshka sparse autoencoders. External Links: 2503.17547, Link Cited by: §D.2, §1, §2.4. D. Chanin, J. Wilken-Smith, T. Dulka, H. Bhatnagar, S. Golechha, and J. Bloom (2025) A is for absorption: studying feature splitting and absorption in sparse autoencoders. External Links: 2409.14507, Link Cited by: §1, §1, §3.5, §3. J. Cui, Q. Zhang, Y. Wang, and Y. Wang (2025) On the theoretical understanding of identifiable sparse autoencoders and beyond. External Links: 2506.15963, Link Cited by: §1. H. Cunningham, A. Ewart, L. Riggs, R. Huben, and L. Sharkey (2023) Sparse autoencoders find highly interpretable features in language models. External Links: 2309.08600, Link Cited by: §D.2, §1, §2.4. D. L. Donoho (2006) Compressed sensing. IEEE Transactions on information theory 52 (4), p. 1289–1306. Cited by: §D.2. J. Dunefsky, P. Chlenski, and N. Nanda (2024) Transcoders find interpretable llm feature circuits. External Links: 2406.11944, Link Cited by: §D.1, §D.2, §1, §2.4. N. Elhage, T. Hume, C. Olsson, N. Schiefer, T. Henighan, S. Kravec, Z. Hatfield-Dodds, R. Lasenby, D. Drain, C. Chen, R. Grosse, S. McCandlish, J. Kaplan, D. Amodei, M. Wattenberg, and C. Olah (2022) Toy models of superposition. External Links: 2209.10652, Link Cited by: §1, §2.2, Assumption 2.4. J. Engels, E. J. Michaud, I. Liao, W. Gurnee, and M. Tegmark (2025) Not all language model features are one-dimensionally linear. External Links: 2405.14860, Link Cited by: 1st item. L. Gao, T. D. la Tour, H. Tillman, G. Goh, R. Troll, A. Radford, I. Sutskever, J. Leike, and J. Wu (2024) Scaling and evaluating sparse autoencoders. External Links: 2406.04093, Link Cited by: §D.2, §1, §1, §2.4. R. Ge, J. D. Lee, and T. Ma (2018) Matrix completion has no spurious local minimum. External Links: 1605.07272, Link Cited by: §D.3. J. Gorski, F. Pfeuffer, and K. Klamroth (2007) Biconvex sets and optimization with biconvex functions: a survey and extensions. Mathematical methods of operations research 66 (3), p. 373–407. Cited by: §D.3, 1st item, 2nd item, §I.5, Example 3.5. R. Gribonval and K. Schnass (2010) Dictionary identification - sparse matrix-factorisation via ℓ1 _1-minimisation. External Links: 0904.4774, Link Cited by: §1. O. Gujral, M. Bafna, E. Alm, and B. Berger (2025) Sparse autoencoders uncover biologically interpretable features in protein language model representations. Proceedings of the National Academy of Sciences 122 (34), p. e2506316122. Cited by: §D.2. P. Jain and P. Kar (2017) Non-convex optimization for machine learning. Foundations and Trends® in Machine Learning 10 (3–4), p. 142–336. External Links: ISSN 1935-8245, Link, Document Cited by: §D.3. Y. Jiang, G. Rajendran, P. Ravikumar, B. Aragam, and V. Veitch (2024) On the origins of linear representations in large language models. External Links: 2403.03867, Link Cited by: §2.2. A. Karvonen, B. Wright, C. Rager, R. Angell, J. Brinkmann, L. Smith, C. M. Verdun, D. Bau, and S. Marks (2024) Measuring progress in dictionary learning for language model interpretability with board game models. External Links: 2408.00113, Link Cited by: §D.2. P. W. Koh, T. Nguyen, Y. S. Tang, S. Mussmann, E. Pierson, B. Kim, and P. Liang (2020) Concept bottleneck models. External Links: 2007.04612, Link Cited by: §D.1. D. D. Lee, P. Pham, Y. Largman, and A. Ng (2009) Advances in neural information processing systems 22. Tech Rep. Cited by: §D.3. J. Lindsey, A. Templeton, J. Marcus, T. Conerly, J. Batson, and C. Olah (2024) Sparse crosscoders for cross-layer features and model diffing. Transformer Circuits Thread. External Links: Link Cited by: §1, §2.4. Z. C. Lipton (2017) The mythos of model interpretability. External Links: 1606.03490, Link Cited by: §1. S. Lundberg and S. Lee (2017) A unified approach to interpreting model predictions. External Links: 1705.07874, Link Cited by: §D.1. Y. Luo, R. An, B. Zou, Y. Tang, J. Liu, and S. Zhang (2024) LLM as dataset analyst: subpopulation structure discovery with large language model. External Links: 2405.02363, Link Cited by: §4.1. J. Mairal, F. Bach, J. Ponce, and G. Sapiro (2009) Online dictionary learning for sparse coding. In Proceedings of the 26th annual international conference on machine learning, p. 689–696. Cited by: 5th item. A. Makhzani and B. Frey (2014) K-sparse autoencoders. External Links: 1312.5663, Link Cited by: §2.4. Z. Mao, J. Xu, Z. Zheng, H. Zheng, D. Sheng, Y. Jin, and G. Yang (2025) Sparse autoencoders bridge the deep learning model and the brain. External Links: 2506.11123, Link Cited by: §D.2. S. Marks, C. Rager, E. J. Michaud, Y. Belinkov, D. Bau, and A. Mueller (2025) Sparse feature circuits: discovering and editing interpretable causal graphs in language models. External Links: 2403.19647, Link Cited by: §1. S. Marks and M. Tegmark (2024) The geometry of truth: emergent linear structure in large language model representations of true/false datasets. External Links: 2310.06824, Link Cited by: §2.2. I. D. Mienye and N. Jere (2024) A survey of decision trees: concepts, algorithms, and applications. IEEE access 12, p. 86716–86727. Cited by: §D.1. N. Nanda, A. Lee, and M. Wattenberg (2023) Emergent linear representations in world models of self-supervised sequence models. External Links: 2309.00941, Link Cited by: §2.2. C. Olah, N. Cammarata, L. Schubert, G. Goh, M. Petrov, and S. Carter (2020) Zoom in: an introduction to circuits. Distill. External Links: Document, Link Cited by: §D.1. C. Olsson, N. Elhage, N. Nanda, N. Joseph, N. DasSarma, T. Henighan, B. Mann, A. Askell, Y. Bai, A. Chen, T. Conerly, D. Drain, D. Ganguli, Z. Hatfield-Dodds, D. Hernandez, S. Johnston, A. Jones, J. Kernion, L. Lovitt, K. Ndousse, D. Amodei, T. Brown, J. Clark, J. Kaplan, S. McCandlish, and C. Olah (2022) In-context learning and induction heads. External Links: 2209.11895, Link Cited by: §D.1. K. Park, Y. J. Choe, Y. Jiang, and V. Veitch (2025) The geometry of categorical and hierarchical concepts in large language models. External Links: 2406.01506, Link Cited by: §2.2. K. Park, Y. J. Choe, and V. Veitch (2024) The linear representation hypothesis and the geometry of large language models. External Links: 2311.03658, Link Cited by: §1, §2.2, Assumption 2.4. G. Paulo, S. Shabalin, and N. Belrose (2025) Transcoders beat sparse autoencoders for interpretability. External Links: 2501.18823, Link Cited by: §2.4. G. Penedo, H. Kydlíček, L. B. allal, A. Lozhkov, M. Mitchell, C. Raffel, L. V. Werra, and T. Wolf (2024) The fineweb datasets: decanting the web for the finest text data at scale. External Links: 2406.17557, Link Cited by: §5.3. A. Radford, J. W. Kim, C. Hallacy, A. Ramesh, G. Goh, S. Agarwal, G. Sastry, A. Askell, P. Mishkin, J. Clark, G. Krueger, and I. Sutskever (2021) Learning transferable visual models from natural language supervision. External Links: 2103.00020, Link Cited by: §5.2. S. Rajamanoharan, A. Conmy, L. Smith, T. Lieberum, V. Varma, J. Kramár, R. Shah, and N. Nanda (2024a) Improving dictionary learning with gated sparse autoencoders. External Links: 2404.16014, Link Cited by: §2.4. S. Rajamanoharan, T. Lieberum, N. Sonnerat, A. Conmy, V. Varma, J. Kramár, and N. Nanda (2024b) Jumping ahead: improving reconstruction fidelity with jumprelu sparse autoencoders. External Links: 2407.14435, Link Cited by: §2.4. C. Rudin (2019) Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead. External Links: 1811.10154, Link Cited by: §1. O. Russakovsky, J. Deng, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, M. Bernstein, A. C. Berg, and L. Fei-Fei (2015) ImageNet large scale visual recognition challenge. External Links: 1409.0575, Link Cited by: §5.2. I. Safran and O. Shamir (2018) Spurious local minima are common in two-layer relu neural networks. External Links: 1712.08968, Link Cited by: §D.2, §D.3. R. R. Selvaraju, M. Cogswell, A. Das, R. Vedantam, D. Parikh, and D. Batra (2019) Grad-cam: visual explanations from deep networks via gradient-based localization. International Journal of Computer Vision 128 (2), p. 336–359. External Links: ISSN 1573-1405, Link, Document Cited by: §D.1. L. Sharkey, B. Chughtai, J. Batson, J. Lindsey, J. Wu, L. Bushnaq, N. Goldowsky-Dill, S. Heimersheim, A. Ortega, J. Bloom, S. Biderman, A. Garriga-Alonso, A. Conmy, N. Nanda, J. Rumbelow, M. Wattenberg, N. Schoots, J. Miller, E. J. Michaud, S. Casper, M. Tegmark, W. Saunders, D. Bau, E. Todd, A. Geiger, M. Geva, J. Hoogland, D. Murfet, and T. McGrath (2025) Open problems in mechanistic interpretability. External Links: 2501.16496, Link Cited by: §D.1. D. Shu, X. Wu, H. Zhao, D. Rai, Z. Yao, N. Liu, and M. Du (2025) A survey on sparse autoencoders: interpreting the internal mechanisms of large language models. External Links: 2503.05613, Link Cited by: §D.1. E. Simon and J. Zou (2024) InterPLM: discovering interpretable features in protein language models via sparse autoencoders. External Links: 2412.12101, Link Cited by: §D.1, §D.2. D. A. Spielman, H. Wang, and J. Wright (2012) Exact recovery of sparsely-used dictionaries. External Links: 1206.5882, Link Cited by: §D.2, §1. R. Sun and Z. Luo (2016) Guaranteed matrix completion via non-convex factorization. IEEE Transactions on Information Theory 62 (11), p. 6535–6579. External Links: ISSN 1557-9654, Link, Document Cited by: §D.3. Y. Tang, A. Lagzian, S. Anumasa, Q. Zou, Y. Zhu, Y. Zhang, T. Nguyen, Y. Tham, E. Adeli, C. Cheng, Y. Du, and D. Liu (2025a) Human-like content analysis for generative ai with language-grounded sparse encoders. External Links: 2508.18236, Link Cited by: §D.1, §D.2. Y. Tang, A. Sinha, and D. Liu (2025b) How does my model fail? automatic identification and interpretation of physical plausibility failure modes with matryoshka transcoders. External Links: 2511.10094, Link Cited by: §D.2, §1, §2.4. A. Templeton, T. Conerly, J. Marcus, J. Lindsey, T. Bricken, B. Chen, A. Pearce, C. Citro, E. Ameisen, A. Jones, H. Cunningham, N. Turner, C. McDougall, M. MacDiarmid, C. D. Freeman, T. R. Sumers, E. Rees, J. Batson, A. Jermyn, S. Carter, C. Olah, and T. Henighan (2024) Scaling monosemanticity: extracting interpretable features from claude 3 sonnet. Transformer Circuits Thread. External Links: Link Cited by: §1, §2.4. V. Visweswaran and C. Floudast (1990) A global optimization algorithm (gop) for certain classes of nonconvex nlps—i. application of theory and test problems. Computers & chemical engineering 14 (12), p. 1419–1434. Cited by: §D.3. M. Wang, T. D. la Tour, O. Watkins, A. Makelov, R. A. Chi, S. Miserendino, J. Wang, A. Rajaram, J. Heidecke, T. Patwardhan, and D. Mossing (2025) Persona features control emergent misalignment. External Links: 2506.19823, Link Cited by: §1. Appendix A Taxonomy for Sparse Dictionary Learning in Mechanistic Interpretability Figure 7: Hierarchical taxonomy of Sparse Dictionary Learning research in Mechanistic Interpretability. Appendix B Notations We summarize the key notations used throughout this paper. Notation Description Distributions and Indexing D Distribution over inputs X Input space s Sample drawn from D de_d Standard basis vector (1 in position d, 0 elsewhere) Representations and Dimensions (s)x(s) Ground-truth features for input s∼s p(s)x_p(s) Input representation to SDL model (observed) r(s)x_r(s) Target/output representation for SDL model q(s)x_q(s) Latent activations in SDL bottleneck n Number of ground-truth features (dimension of x) npn_p Dimension of input representation px_p nrn_r Dimension of target representation rx_r nqn_q Dimension of latent space qx_q SDL Architecture WEW_E Encoder matrix (nq×npn_q× n_p) WDW_D Decoder matrix (nr×nqn_r× n_q) σ(⋅)σ(·) Sparsity-inducing activation function Eiw_E^i i-th row of encoder WEW_E (encodes to neuron i) Diw_D^i i-th column of decoder WDW_D (decodes from neuron i) Linear Representation Hypothesis WpW_p Feature matrix for px_p (np×n_p× n) WrW_r Feature matrix for rx_r (nr×n_r× n) pdw_p^d d-th column of WpW_p (feature direction for feature d) rdw_r^d d-th column of WrW_r (feature direction for feature d) WptrueW_p^true Ground-truth feature matrix (Linear Representation Bench) WplearnedW_p^learned Learned feature matrix (WE⊤W_E ) Sparsity and Interference S Sparsity level: Pr⁡(i(s)=0)≥S (x_i(s)=0)≥ S M Maximum interference: maxi≠j⁡⟨Wp[:,i],Wp[:,j]⟩ _i≠ j W_p[:,i],W_p[:,j] MdM_d Weight for feature d: Pr⁡((s)=xd(s)d)⋅[xd(s)2|xd(s)>0] (x(s)=x_d(s)e_d)·E[x_d(s)^2|x_d(s)>0] Feature Anchoring k Number of anchor features ~p(i),~r(i) w_p^(i), w_r^(i) i-th anchor feature pair λanchor _anchor Anchoring loss weight K Set of indices for selected anchor features Loss Functions ℒSDLL_SDL Standard SDL reconstruction loss ℒ~SDL L_SDL Approximate SDL loss (extreme sparsity regime) ℒanchorL_anchor Feature anchoring penalty ℒSDL-FAL_SDL-FA Anchored SDL objective: ℒSDL+λanchorℒanchorL_SDL+ _anchorL_anchor Activation Patterns =(ℱ1,…,ℱN)P=(F_1,…,F_N) Activation pattern ℱiF_i Set of ground-truth features activating neuron i (d)A(d) Set of neurons activated by feature d Ω _A Activation pattern region (fixed activation structure) c Activation threshold for σ Evaluation Metrics GTM_GT GT Recovery: fraction of features with >τ>τ alignment IPM_IP Maximum Inner Product: mean best alignment per feature τ Threshold for GT Recovery (typically τ=0.9τ=0.9 or 0.950.95) S Similarity matrix: |Wplearned⊤Wptrue||W_p^learned W_p^true| Appendix C The Linear Representation Bench The Linear Representation Bench is a synthetic benchmark designed to precisely instantiate the Linear Representation Hypothesis (Assumption 2.3) and Representation Assumptions (Assumption 2.4) with fully accessible ground-truth features (See Figure 8 for a visualization of the benchmark). This enables rigorous evaluation of SDL methods under controlled conditions where feature recovery can be directly measured. Figure 8: Visualization of the Linear Representation Bench. The figure illustrates a D=3D=3 dimensional representation space generated using N=4N=4 feature directions (pW_p). C.1 Ground-Truth Feature Matrix Generation We construct the feature matrix p∈ℝnp×nW_p ^n_p× n to satisfy the unit-norm and bounded interference conditions (Assumption 2.4). Initialization. Initialize pW_p with random Gaussian entries and normalize each column to unit ℓ2 _2-norm: pd←pd‖pd‖2,∀d∈[n]w_p^d← w_p^d\|w_p^d\|_2, ∀ d∈[n] (26) Interference Minimization. We minimize pairwise interference via projected gradient descent. Define the soft-thresholded interference loss: ℒint(p)=∑i≠j[max⁡(0,⟨pi,pj⟩−(M−ϵ))]2+λ∑i≠j[max⁡(0,⟨pi,pj⟩)]2L_int(W_p)= _i≠ j [ (0, _p^i,w_p^j -(M-ε) ) ]^2+λ _i≠ j [ (0, _p^i,w_p^j ) ]^2 (27) where ϵ>0ε>0 is a tolerance margin and λ>0λ>0 is a small regularization weight encouraging negative interference. At each iteration, we compute the gradient with respect to pW_p, perform a gradient step, and project back to the unit sphere: p←p−η∇pℒint,pd←pd‖pd‖2,∀d∈[n]W_p _p-η _W_pL_int, _p^d← w_p^d\|w_p^d\|_2, ∀ d∈[n] (28) This procedure continues until the maximum interference satisfies maxi≠j⁡⟨pi,pj⟩≤M _i≠ j _p^i,w_p^j ≤ M. C.2 Sparse Coefficient Generation For each sample i∈[N]i∈[N], we generate sparse ground-truth features (i)∈ℝ+nx^(i) ^n_+ as follows: Sparsity Mask. Each feature d∈[n]d∈[n] is independently activated with probability (1−)(1-S): md(i)∼Bernoulli(1−)m_d^(i) (1-S) (29) Feature Magnitudes. Active features follow a shifted exponential distribution: xd(i)=md(i)⋅(cmin+Exp(β))x_d^(i)=m_d^(i)· (c_ +Exp(β) ) (30) where cmin≥0c_ ≥ 0 is a minimum activation threshold and β>0β>0 is the scale parameter. This construction ensures: • Non-negativity: xd(i)≥0x_d^(i)≥ 0 for all d,id,i • Sparsity: Pr⁡(xd(i)=0)= (x_d^(i)=0)=S for all d • Independence: Features xd(i)d=1n\x_d^(i)\_d=1^n are mutually independent C.3 Data Synthesis The observed representations are computed via linear combination: p(i)=p(i)=∑d=1nxd(i)pdx_p^(i)=W_px^(i)= _d=1^nx_d^(i)w_p^d (31) By construction, this dataset exactly satisfies Assumptions 2.3 and 2.4, enabling evaluation of SDL methods with complete knowledge of ground-truth features. C.4 Default Configuration Table 3 lists the default parameters used in our experiments. Table 3: Default configuration for the Linear Representation Bench. Parameter Symbol Default Value Number of features n 1000 Representation dimension npn_p 768 Superposition ratio n/npn/n_p 1.30× Number of samples N 100,000 Sparsity level S 0.99 Maximum interference M 0.1 This controlled setting enables direct measurement of feature recovery metrics with ground-truth access, complementing evaluations on real neural network representations where true features are unknown. Appendix D Related Works D.1 Mechanistic Interpretability Interpretability is crucial for deploying AI in high-stakes domains such as medical diagnosis and financial modeling, where understanding model decisions is essential for safety and trust (Simon and Zou, 2024; Abdulaal et al., 2024). Traditional approaches include interpretability-by-design methods like Concept Bottleneck Models (Koh et al., 2020) and decision trees (Mienye and Jere, 2024), and post-hoc explanation methods like GradCAM (Selvaraju et al., 2019) and SHAP (Lundberg and Lee, 2017). Mechanistic interpretability (Sharkey et al., 2025; Bereska and Gavves, 2024) aims to reverse-engineer neural networks by understanding their internal computational mechanisms. SAEs (Shu et al., 2025) and related dictionary learning methods (Tang et al., 2025a; Dunefsky et al., 2024) decompose neural activations into sparse, interpretable features. Circuit analysis (Olah et al., 2020; Olsson et al., 2022) investigates how these features compose into computational algorithms. D.2 Sparse Dictionary Learning Sparse dictionary learning has a rich history predating its application to mechanistic interpretability. K-SVD (Aharon et al., 2006) established foundational methods for learning overcomplete dictionaries, while theoretical work in compressed sensing (Donoho, 2006) characterized recovery conditions, with Spielman et al. (2012) providing polynomial-time algorithms for exact reconstruction under sparsity assumptions. Safran and Shamir (2018) demonstrated that spurious local minima are common even in simple two-layer ReLU networks, highlighting optimization challenges that persist in modern applications. Recent work has adapted these principles to mechanistic interpretability. SAEs (Cunningham et al., 2023) apply dictionary learning to language model activations. Various variants are further developed, including transcoders (Dunefsky et al., 2024), crosscoders (Gao et al., 2024), Matryoshka SAEs (Bussmann et al., 2025), and hybrid approaches like Language-Grounded Sparse Encoders (Tang et al., 2025a). These methods have found applications beyond language models, including protein structure analysis (Simon and Zou, 2024; Gujral et al., 2025), medical imaging (Abdulaal et al., 2024), model evaluation (Tang et al., 2025b), board game analysis (Karvonen et al., 2024), and fMRI data analysis Mao et al. (2025). D.3 Biconvex Optimization Biconvex optimization studies problems where the objective is convex in each variable block when the other is fixed. Gorski et al. (2007) provide a comprehensive survey establishing theoretical foundations and algorithmic approaches, while Visweswaran and Floudast (1990) develop the GOP algorithm providing global optimality guarantees via branch-and-bound. Matrix factorization problems exhibit similar bilinear structure; Lee et al. (2009) introduce non-negative matrix factorization with multiplicative updates, and subsequent work establishes landscape properties, with Ge et al. (2018) proving matrix completion has no spurious local minima and Sun and Luo (2016) providing convergence guarantees for alternating minimization. More broadly, Jain and Kar (2017) survey non-convex optimization in machine learning, while Safran and Shamir (2018) demonstrate that spurious local minima are common in two-layer ReLU networks. Our work bridges these optimization-theoretic foundations with mechanistic interpretability by proving SDL exhibits biconvex structure, enabling the application of established algorithms and analysis techniques to this emerging field. Appendix E Limitations While our theoretical framework provides valuable insights into SDL methods, several limitations warrant discussion: • Assumption Violations. Our analysis relies on Assumptions 2.3 and 2.4, which may not hold perfectly in real-world neural networks. In particular, there exists features that are not one-dimentional linear (Engels et al., 2025) . • Extreme Sparsity. Several key results, including Theorem 3.1, rely on the extreme sparsity assumption (S→1S→ 1). The bounds may degrade substantially for moderate sparsity levels commonly observed in practice. • Feature Independence. We assume mutual independence among ground-truth features, but real-world concepts often exhibit correlations that our analysis does not capture. • Convergence Guarantees. We characterize global and partial minima of the optimization landscape but do not provide guarantees on whether gradient-based methods converge to global minima. • Anchor Availability. Feature anchoring requires access to known semantic directions, which may not always be available. The quality and coverage of anchors significantly impact performance. Appendix F Future Works Our theoretical framework opens several avenues for future research: • Global Optimization via GOP. Having established that SDL is piecewise biconvex (Theorem 3.2), applying Global Optimization for biconvex Problems (GOP) (Gorski et al., 2007) could provide certificates of global optimality and systematically escape spurious partial minima. • Alternating Convex Search. Alternating convex search (ACS) (Gorski et al., 2007) directly exploits biconvex structure by alternately solving convex subproblems for WDW_D and WEW_E. This may offer faster convergence and better solutions than standard gradient descent. • Convergence and Sample Complexity. Building on our landscape characterization, future work should establish convergence rates for gradient descent and sample complexity bounds for feature recovery. • Moderate Sparsity Analysis. Extending results beyond the S→1S→ 1 regime to handle feature co-activation would broaden practical applicability. • Classical Dictionary Learning. Adapting established algorithms like K-SVD (Aharon et al., 2006) and online dictionary learning (Mairal et al., 2009) to the SDL setting could yield improved optimization methods. • Automatic Anchor Discovery. Developing methods to automatically discover high-quality anchors without external supervision would make feature anchoring more practical. Appendix G Ablation Studies Our theoretical analysis reveals that SDL optimization becomes increasingly underdetermined as the number of ground-truth features n grows relative to the representation dimension npn_p (Theorem 3.4). To comprehensively validate our theoretical framework and assess the robustness of feature anchoring across different conditions, we conduct extensive ablation studies on the Linear Representation Bench, systematically varying superposition ratio, interference level, sparsity, and activation function parameters. Effect of Superposition Ratio. We first examine how feature anchoring performs under varying degrees of superposition by training ReLU SAEs with nq=16,000n_q=16,000 latent dimensions while varying the number of ground-truth features n∈800,900,1000,1100,1200,1300,1400n∈\800,900,1000,1100,1200,1300,1400\. The representation dimension is fixed at np=nr=768n_p=n_r=768, yielding superposition ratios from 1.04×1.04× to 1.82×1.82×. For feature anchoring, we randomly select k=100k=100 ground-truth features as anchors with λanchor=0.1 _anchor=0.1. As shown in Table 4, standard SAE training achieves 0% GT Recovery across all feature counts, demonstrating the severity of the underdetermined optimization problem. With feature anchoring, GT Recovery reaches 100% at n=800n=800 and remains high (98.89%) at n=900n=900, confirming that anchoring a small subset of features (12.5% and 11.1% respectively) provides sufficient constraint to guide optimization toward the global minimum. As superposition increases, GT Recovery gradually decreases (85.50% at 1.30×1.30×, 58.09% at 1.43×1.43×, 30.58% at 1.56×1.56×), aligning with our theoretical expectation that higher superposition expands the space of spurious solutions. Notably, feature anchoring consistently improves Maximum Inner Product across all settings, indicating better feature quality even when full recovery is not achieved. Table 4: Feature recovery under varying superposition ratios. We fix np=nr=768n_p=n_r=768 and vary n from 800 to 1400. Feature anchoring uses k=100k=100 randomly selected anchors with λanchor=0.1 _anchor=0.1. Method Num Features GT Recovery ↑ Max Inner Product ↑ SAE 800 0.00% 0.361 + Feature Anchoring 800 100.00% 0.436 SAE 900 0.00% 0.298 + Feature Anchoring 900 98.89% 0.345 SAE 1000 0.00% 0.254 + Feature Anchoring 1000 85.50% 0.292 SAE 1100 0.00% 0.223 + Feature Anchoring 1100 58.09% 0.258 SAE 1200 0.00% 0.203 + Feature Anchoring 1200 30.58% 0.233 SAE 1300 0.00% 0.192 + Feature Anchoring 1300 15.23% 0.216 SAE 1400 0.00% 0.181 + Feature Anchoring 1400 8.57% 0.207 Effect of Maximum Interference. To examine how feature anchoring performs under different interference levels, we fix n=1000n=1000 features in np=nr=768n_p=n_r=768 dimensions and vary the maximum interference M∈0.05,0.1,0.2,0.5M∈\0.05,0.1,0.2,0.5\ during ground-truth feature matrix generation. Recall from Assumption 2.4 that M:=maxi≠j⁡⟨Wp[:,i],Wp[:,j]⟩M:= _i≠ j W_p[:,i],W_p[:,j] quantifies the maximum dot product between distinct feature directions, characterizing the degree of feature overlap in the representation space. Table 5 presents results across different interference levels. For TopK, BatchTopK, and Matryoshka SAEs, feature anchoring consistently improves GT Recovery across all interference values, with improvements ranging from 2-7 percentage points. For instance, at M=0.5M=0.5, BatchTopK SAE improves from 83.7% to 89.6% with anchoring. ReLU and JumpReLU SAEs, which struggle with feature recovery even with anchoring (0% GT Recovery), still show improved Maximum Inner Product (e.g., ReLU at M=0.2M=0.2: 0.204 → 0.245), indicating better feature alignment. Interestingly, performance degrades slightly at very low interference (M=0.05M=0.05) compared to moderate interference (M=0.1,0.2M=0.1,0.2), likely because extremely low interference creates near-orthogonal features that are easier to recover without additional constraints, making the anchoring less critical. Table 5: Feature recovery under varying maximum interference M. All experiments use n=1000n=1000 features with np=nr=768n_p=n_r=768 and k=100k=100 anchors. Method Max Interference GT Recovery ↑ Max Inner Product ↑ Low Interference (M=0.05M=0.05) ReLU SAE 0.05 0.0% 0.205 + Feature Anchoring 0.05 0.0% 0.246 JumpReLU SAE 0.05 0.0% 0.239 + Feature Anchoring 0.05 0.0% 0.326 TopK SAE 0.05 84.3% 0.982 + Feature Anchoring 0.05 87.3% 0.985 BatchTopK SAE 0.05 86.1% 0.984 + Feature Anchoring 0.05 87.5% 0.986 Matryoshka SAE 0.05 83.8% 0.982 + Feature Anchoring 0.05 85.8% 0.985 Moderate Interference (M=0.2M=0.2) ReLU SAE 0.2 0.0% 0.204 + Feature Anchoring 0.2 0.0% 0.245 JumpReLU SAE 0.2 0.0% 0.237 + Feature Anchoring 0.2 0.0% 0.333 TopK SAE 0.2 85.2% 0.984 + Feature Anchoring 0.2 86.8% 0.986 BatchTopK SAE 0.2 84.7% 0.981 + Feature Anchoring 0.2 90.7% 0.989 Matryoshka SAE 0.2 84.7% 0.983 + Feature Anchoring 0.2 87.6% 0.986 High Interference (M=0.5M=0.5) ReLU SAE 0.5 0.0% 0.203 + Feature Anchoring 0.5 0.0% 0.243 JumpReLU SAE 0.5 0.0% 0.237 + Feature Anchoring 0.5 0.0% 0.325 TopK SAE 0.5 84.2% 0.982 + Feature Anchoring 0.5 88.2% 0.987 BatchTopK SAE 0.5 83.7% 0.981 + Feature Anchoring 0.5 89.6% 0.987 Matryoshka SAE 0.5 83.9% 0.982 + Feature Anchoring 0.5 86.4% 0.985 Effect of Feature Sparsity. We investigate how feature sparsity S affects SDL performance by varying S∈0.005,0.01,0.05,0.1S∈\0.005,0.01,0.05,0.1\ where S denotes the probability that each feature is inactive (Assumption 2.3). Lower S values correspond to denser activation patterns, while higher S approaches the extreme sparsity regime analyzed in our theoretical results. Table 6 reveals several key insights. At very low sparsity (S=0.005S=0.005), where features co-activate frequently, feature anchoring provides the most dramatic improvements. JumpReLU SAE improves from 11.8% to 74.8% GT Recovery, and all TopK-family methods show 2-6 percentage point gains. This validates that anchoring is particularly valuable when the extreme sparsity assumption is violated and feature co-occurrence complicates optimization. At moderate sparsity (S=0.05S=0.05), performance drops significantly across all methods, as increased co-activation creates more complex interference patterns. At high sparsity (S=0.1S=0.1), approaching our theoretical regime, TopK-family methods achieve near-perfect recovery (98.8%-99.7%), with anchoring providing marginal improvements. Interestingly, Matryoshka SAE benefits least from anchoring at extreme sparsity, likely because its multi-scale architecture already provides sufficient constraints to recover features. Table 6: Feature recovery under varying sparsity levels S. All experiments use n=1000n=1000 features with np=nr=768n_p=n_r=768 and k=100k=100 anchors. Method Sparsity GT Recovery ↑ Max Inner Product ↑ Very Low Sparsity (S=0.005S=0.005) ReLU SAE 0.005 0.0% 0.231 + Feature Anchoring 0.005 0.0% 0.326 JumpReLU SAE 0.005 11.8% 0.890 + Feature Anchoring 0.005 74.8% 0.970 TopK SAE 0.005 82.2% 0.978 + Feature Anchoring 0.005 84.3% 0.981 BatchTopK SAE 0.005 80.5% 0.976 + Feature Anchoring 0.005 83.2% 0.980 Matryoshka SAE 0.005 79.3% 0.974 + Feature Anchoring 0.005 86.1% 0.983 Moderate Sparsity (S=0.01S=0.01) ReLU SAE 0.01 0.00% 0.205 + Feature Anchoring 0.01 0.00% 0.246 JumpReLU SAE 0.01 0.00% 0.237 + Feature Anchoring 0.01 0.00% 0.333 TopK SAE 0.01 84.90% 0.983 + Feature Anchoring 0.01 87.63% 0.986 BatchTopK SAE 0.01 84.80% 0.981 + Feature Anchoring 0.01 89.38% 0.988 Matryoshka SAE 0.01 83.70% 0.982 + Feature Anchoring 0.01 87.32% 0.985 High Sparsity (S=0.1S=0.1) ReLU SAE 0.1 0.0% 0.186 + Feature Anchoring 0.1 0.0% 0.191 JumpReLU SAE 0.1 0.0% 0.191 + Feature Anchoring 0.1 0.0% 0.193 TopK SAE 0.1 98.8% 0.981 + Feature Anchoring 0.1 98.4% 0.979 BatchTopK SAE 0.1 95.6% 0.975 + Feature Anchoring 0.1 96.2% 0.976 Matryoshka SAE 0.1 99.7% 0.991 + Feature Anchoring 0.1 99.7% 0.991 Effect of TopK Sparsity Parameter. Finally, we examine how the sparsity-inducing parameter k in TopK activation affects feature recovery. We train TopK SAEs with k∈32,64,128k∈\32,64,128\, corresponding to average activation rates of 0.2%, 0.4%, and 0.8% of the nq=16,000n_q=16,000 latent dimensions. As shown in Table 7, both standard and anchored TopK SAEs achieve perfect (100%) GT Recovery at k=32k=32, where extreme sparsity closely matches our theoretical assumptions. At k=64k=64, GT Recovery remains high (93.4%-93.9%) with minimal difference between standard and anchored training, indicating that moderate sparsity provides sufficient constraint for feature recovery without additional anchoring. At k=128k=128, where activation density increases, feature anchoring becomes beneficial again, improving GT Recovery from 86.7% to 88.0%. The Maximum Inner Product remains consistently high (>0.985>0.985) across all settings, confirming that TopK activation is generally effective for feature recovery, with anchoring providing incremental benefits at higher k values where the optimization becomes more challenging. Table 7: Feature recovery under varying TopK sparsity parameter k. All experiments use n=1000n=1000 features with np=nr=768n_p=n_r=768, nq=16,000n_q=16,000, and k=100k=100 anchors. Method TopK Parameter k GT Recovery ↑ Max Inner Product ↑ TopK SAE 32 100.0% 0.999 + Feature Anchoring 32 100.0% 0.999 TopK SAE 64 93.9% 0.992 + Feature Anchoring 64 93.4% 0.991 TopK SAE 128 86.7% 0.986 + Feature Anchoring 128 88.0% 0.987 Theoretical Interpretation. These comprehensive ablation studies validate our theoretical framework across diverse conditions. The consistent failure of ReLU and JumpReLU SAEs without sufficient sparsity constraints corroborates Theorem 3.4: the underdetermined solution space admits infinitely many configurations achieving low reconstruction loss without recovering interpretable features. Feature anchoring addresses this by constraining encoder-decoder pairs, reducing degrees of freedom and steering optimization away from spurious partial minima (Theorem 3.7). The varying effectiveness across interference levels, sparsity regimes, and activation functions reflects the fundamental trade-off in Assumption 2.4: more severe feature compression (higher M), denser activations (lower S), or insufficient activation sparsity (higher k) create more complex optimization landscapes where even anchored methods struggle to disentangle all features completely. Appendix H Qualitative Examples Beyond quantitative metrics, we provide qualitative evidence demonstrating how feature anchoring improves feature monosemanticity. Figure 9 shows features learned by Matryoshka SAE with feature anchoring on CLIP embeddings of ImageNet, while Figure 10 shows features from the same architecture trained without feature anchoring. With Feature Anchoring. As shown in Figure 9, the learned features exhibit clear monosemanticity: each feature responds to a single, well-defined visual concept. The “African Grey” feature activates exclusively on African Grey parrots, the “American Black Bear” feature captures only black bears, and the “Digital Clock” feature responds specifically to digital time displays. This monosemantic behavior aligns with our theoretical prediction that feature anchoring reduces the underdetermined nature of SDL optimization (Theorem 3.4). Without Feature Anchoring. In contrast, Figure 10 illustrates the polysemanticity that emerges without anchoring. Features respond to multiple unrelated concepts: one activates on “Various Boxes” (dishwashers, file cabinets, chests), another on “Various Screens” (slot machines, scoreboards, televisions). While achieving low reconstruction loss, these features fail to capture semantically meaningful concepts—exactly the spurious partial minima characterized in Theorem 3.7. Figure 9: Features learned with feature anchoring exhibit monosemanticity. Each row shows the top-activating images for a single feature from Matryoshka SAE trained on CLIP embeddings with feature anchoring. Figure 10: Features learned without feature anchoring exhibit polysemanticity. Each row shows the top-activating images for a single feature from Matryoshka SAE trained without feature anchoring. Appendix I Proofs of the Theorems I.1 Proof of Theorem 3.1 Proof. We establish how the loss decomposes when S→1S→ 1. Let Wr=[wr1,wr2,…,wrn]W_r=[w_r^1,w_r^2,…,w_r^n] and Wp=[wp1,wp2,…,wpn]W_p=[w_p^1,w_p^2,…,w_p^n] denote the column representations of the reconstruction and projection matrices. Under extreme sparsity (S→1S→ 1), we only need to consider situations where at most one component xd(s)x_d(s) is active for any given input. The expectation decomposes over sparsity levels: ℒSDL _SDL =s‖Wr(s)−WDσ(WEWp(s))‖2 =E_s \|W_rx(s)-W_Dσ(W_EW_px(s)) \|^2 (32) =∑m=0nPr(∥(s)∥0=m)⋅[∥r(s)−WDσ(WEp(s))∥2∣∥(s)∥0=m] = _m=0^n (\|x(s)\|_0=m)·E[\|x_r(s)-W_Dσ(W_Ex_p(s))\|^2 \|x(s)\|_0=m] (33) When ‖(s)‖0=0\|x(s)\|_0=0, both r(s)=0x_r(s)=0 and the reconstruction WDσ(WEp(s))=0W_Dσ(W_Ex_p(s))=0, contributing zero to the loss. When ‖(s)‖0=1\|x(s)\|_0=1, exactly one feature is active: (s)=xd(s)dx(s)=x_d(s)e_d for some d∈[n]d∈[n]. In this case: r(s) _r(s) =Wr(s)=xd(s)wrd =W_rx(s)=x_d(s)w_r^d (34) p(s) _p(s) =Wp(s)=xd(s)wpd =W_px(s)=x_d(s)w_p^d (35) Therefore: ∑d=1nPr⁡((s)=xd(s)d)⋅[‖r(s)−WDσ(WEp(s))‖2∣(s)=xd(s)d] _d=1^n (x(s)=x_d(s)e_d)·E[\|x_r(s)-W_Dσ(W_Ex_p(s))\|^2 (s)=x_d(s)e_d] (36) =∑d=1nPr⁡((s)=xd(s)d)⋅xd(s)[xd(s)2‖wrd−WDσ(WEwpd)‖2∣xd(s)>0] = _d=1^n (x(s)=x_d(s)e_d)·E_x_d(s) [x_d(s)^2 \|w_r^d-W_Dσ(W_Ew_p^d) \|^2 x_d(s)>0 ] (37) =∑d=1nMd‖wrd−WDσ(WEwpd)‖2 = _d=1^nM_d \|w_r^d-W_Dσ(W_Ew_p^d) \|^2 (38) where Md:=Pr⁡((s)=xd(s)d)⋅[xd(s)2|xd(s)>0]M_d:= (x(s)=x_d(s)e_d)·E[x_d(s)^2|x_d(s)>0]. When ‖(s)‖0≥2\|x(s)\|_0≥ 2, by Assumption 2.4 (extreme sparsity), ∑m=2nPr⁡(‖(s)‖0=m)≤(1−S)2 _m=2^n (\|x(s)\|_0=m)≤(1-S)^2. Since the reconstruction error is bounded by a constant C (due to bounded norms), this contribution is O((1−S)2)O((1-S)^2). Combining all terms: ℒSDL=∑d=1nMd‖wrd−WDσ(WEwpd)‖2+O((1−S)2)L_SDL= _d=1^nM_d \|w_r^d-W_Dσ(W_Ew_p^d) \|^2+O((1-S)^2) (39) ∎ I.2 Proof of Theorem 3.2 Proof. Notation. Recall that (d)⊆[nq]A(d) [n_q] denotes the set of neurons activated when feature d is active in isolation. Formally, for WE∈ΩW_E∈ _A: (d)=i∈[nq]:⟨wEi,wpd⟩>cA(d)=\i∈[n_q]: w_E^i,w_p^d >c\ (40) where c≥0c≥ 0 is the activation threshold. Preliminary observation. Within Ω _A, the activation zd(WE):=σ(WEwpd)z_d(W_E):=σ(W_Ew_p^d) satisfies: zd(WE)i=⟨wEi,wpd⟩if i∈(d)0if i∉(d)z_d(W_E)^i= cases w_E^i,w_p^d &if i (d)\\ 0&if i (d) cases (41) Therefore, zd(WE)z_d(W_E) is a linear function of WEW_E within Ω _A. Per-feature loss decomposition. Define the per-feature reconstruction loss: fd(WD,WE):=‖wrd−WDσ(WEwpd)‖2f_d(W_D,W_E):=\|w_r^d-W_Dσ(W_Ew_p^d)\|^2 (42) The approximate SDL loss is a non-negative weighted sum: ℒ~SDL(WD,WE)=∑d=1nMdfd(WD,WE) L_SDL(W_D,W_E)= _d=1^nM_df_d(W_D,W_E) (43) where Md>0M_d>0 for all d∈[n]d∈[n]. We prove biconvexity by establishing that each fdf_d is biconvex, from which the result follows since positive linear combinations preserve biconvexity. Part 1: fdf_d is convex in WDW_D for fixed WE∈ΩW_E∈ _A. Fix WE∈ΩW_E∈ _A and let zd:=zd(WE)z_d:=z_d(W_E) be the fixed activation vector. Then: fd(WD)=‖wrd−WDzd‖2f_d(W_D)=\|w_r^d-W_Dz_d\|^2 (44) Gradient. The gradient with respect to WDW_D is: ∇WDfd=−2(wrd−WDzd)zd⊤ _W_Df_d=-2(w_r^d-W_Dz_d)z_d (45) Hessian. The Hessian (treating WDW_D as vectorized) is: ∇WD2fd=2Inr⊗(zdzd⊤)∇^2_W_Df_d=2I_n_r (z_dz_d ) (46) Since zdzd⊤⪰0z_dz_d 0, the Kronecker product with InrI_n_r is PSD, confirming fdf_d is convex in WDW_D. Part 2: fdf_d is convex in WEW_E for fixed WDW_D, within Ω _A. Fix WD∈ℝnr×nqW_D ^n_r× n_q. Within Ω _A, using the linearity of zd(WE)z_d(W_E): WDzd(WE)=∑i∈(d)⟨wEi,wpd⟩wDi=:ud(WE)W_Dz_d(W_E)= _i (d) w_E^i,w_p^d w_D^i=:u_d(W_E) (47) where ud(WE)u_d(W_E) is linear (hence affine) in WEW_E. Thus: fd(WE)=‖wrd−ud(WE)‖2f_d(W_E)=\|w_r^d-u_d(W_E)\|^2 (48) Since the squared norm of an affine function is convex, fdf_d is convex in WEW_E. Verification via Hessian. For each encoder row wEiw_E^i with i∈(d)i (d), the gradient is: ∇wEifd=−2(wrd−ud(WE))⊤wDi⋅wpd _w_E^if_d=-2(w_r^d-u_d(W_E)) w_D^i· w_p^d (49) The Hessian is: ∇wEi2fd=2‖wDi‖2(wpd(wpd)⊤)∇^2_w_E^if_d=2\|w_D^i\|^2(w_p^d(w_p^d) ) (50) Since wpd(wpd)⊤⪰0w_p^d(w_p^d) 0 and ‖wDi‖2≥0\|w_D^i\|^2≥ 0, we have ∇wEi2fd⪰0∇^2_w_E^if_d 0. For i∉(d)i (d), neuron i does not affect fdf_d, so ∇wEi2fd=0∇^2_w_E^if_d=0. The full Hessian ∇WE2fd∇^2_W_Ef_d is block-diagonal with PSD blocks, hence ∇WE2fd⪰0∇^2_W_Ef_d 0. Conclusion. Since each fdf_d is biconvex and Md>0M_d>0 for all d, the weighted sum: ℒ~SDL(WD,WE)=∑d=1nMdfd(WD,WE) L_SDL(W_D,W_E)= _d=1^nM_df_d(W_D,W_E) (51) is biconvex over ℝnr×nq×ΩR^n_r× n_q× _A, as positive linear combinations preserve biconvexity. ∎ I.3 Proof of Theorem 3.3 Proof. We analyze the approximate loss ℒ~SDL(WD∗,WE∗) L_SDL(W_D^*,W_E^*) by examining the reconstruction error for each ground-truth feature when active in isolation. Consider feature d∈[n]d∈[n] active in isolation. The encoder output is WE∗wpd=Wp⊤wpdW_E^*w_p^d=W_p w_p^d with k-th component: (WE∗wpd)k=⟨wpk,wpd⟩(W_E^*w_p^d)_k= w_p^k,w_p^d (52) Under Assumption 2.4, the unit-norm condition gives (WE∗wpd)d=1(W_E^*w_p^d)_d=1. For k≠dk≠ d, the bounded interference condition implies ⟨wpk,wpd⟩≤M w_p^k,w_p^d ≤ M, where M is the maximum interference. Combined with the Cauchy-Schwarz inequality, we have: ⟨wpk,wpd⟩∈[−1,M]∀k≠d w_p^k,w_p^d ∈[-1,M] ∀ k≠ d (53) Define the set of activated neurons for feature d: d=k∈[n]:σ(WE∗wpd)k>0A_d=\k∈[n]:σ(W_E^*w_p^d)_k>0\ (54) Since σ(z)i∈0,ziσ(z)_i∈\0,z_i\ and typical activation functions satisfy σ(z)i>0⇔zi>cσ(z)_i>0 z_i>c for some threshold c≥0c≥ 0, we have: d=k∈[n]:⟨wpk,wpd⟩>cA_d=\k∈[n]: w_p^k,w_p^d >c\ (55) Note that d∈d _d since (WE∗wpd)d=1>c≥0(W_E^*w_p^d)_d=1>c≥ 0. For k≠dk≠ d, neuron k activates only if ⟨wpk,wpd⟩>c≥0 w_p^k,w_p^d >c≥ 0, which requires positive interference. Therefore: d∖d⊆k∈[n]:0<⟨wpk,wpd⟩≤MA_d \d\ \k∈[n]:0< w_p^k,w_p^d ≤ M\ (56) Let Kd=|d|K_d=|A_d| denote the number of activated neurons. The activation output satisfies σ(WE∗wpd)k=⟨wpk,wpd⟩σ(W_E^*w_p^d)_k= w_p^k,w_p^d for k∈dk _d and zero otherwise, since σ(z)i∈0,ziσ(z)_i∈\0,z_i\. The reconstruction is therefore WD∗σ(WE∗wpd)=∑k∈d⟨wpk,wpd⟩wrkW_D^*σ(W_E^*w_p^d)= _k _d w_p^k,w_p^d w_r^k. The reconstruction error for feature d is wrd−WD∗σ(WE∗wpd) w_r^d-W_D^*σ(W_E^*w_p^d) =wrd−∑k∈d⟨wpk,wpd⟩wrk =w_r^d- _k _d w_p^k,w_p^d w_r^k (57) =wrd−wrd−∑k∈d∖d⟨wpk,wpd⟩wrk =w_r^d-w_r^d- _k _d \d\ w_p^k,w_p^d w_r^k (58) =−∑k∈d∖d⟨wpk,wpd⟩wrk =- _k _d \d\ w_p^k,w_p^d w_r^k (59) Under the unit-norm assumption, ‖wrk‖=1\|w_r^k\|=1 for all k. For k∈d∖dk _d \d\, we have ⟨wpk,wpd⟩>0 w_p^k,w_p^d >0 (since negative interference does not activate) and ⟨wpk,wpd⟩≤M w_p^k,w_p^d ≤ M by the bounded interference condition. Therefore, using the triangle inequality: ‖wrd−WD∗σ(WE∗wpd)‖ \|w_r^d-W_D^*σ(W_E^*w_p^d) \| ≤∑k∈d∖d|⟨wpk,wpd⟩|⋅‖wrk‖ ≤ _k _d \d\| w_p^k,w_p^d |·\|w_r^k\| (60) =∑k∈d∖d⟨wpk,wpd⟩⋅1(positive terms) = _k _d \d\ w_p^k,w_p^d · 1 (positive terms) (61) ≤∑k∈d∖dM ≤ _k _d \d\M (62) =M(Kd−1) =M(K_d-1) (63) Squaring both sides: ‖wrd−WD∗σ(WE∗wpd)‖2≤M2(Kd−1)2\|w_r^d-W_D^*σ(W_E^*w_p^d)\|^2≤ M^2(K_d-1)^2. The approximate loss is ℒ~SDL(WD∗,WE∗)=∑d=1nMd⋅M2(Kd−1)2≤M2∑d=1nMd(Kd−1)2 L_SDL(W_D^*,W_E^*)= _d=1^nM_d· M^2(K_d-1)^2≤ M^2 _d=1^nM_d(K_d-1)^2 (64) Since Kd≥1K_d≥ 1 for all d and typically Kd=O(1)K_d=O(1) for most features (as the activation function provides sparsity), we have (Kd−1)2=O(1)(K_d-1)^2=O(1). Under the feature independence and stability assumptions (Assumption 2.4), ∑d=1nMd=∑d=1nPr⁡((s)=xd(s)ed)⋅[xd(s)2|xd(s)>0]=O(1−S) _d=1^nM_d= _d=1^n (x(s)=x_d(s)e_d)·E[x_d(s)^2|x_d(s)>0]=O(1-S), since the probability that exactly one feature is active is (1−S)(1-S) under extreme sparsity. Therefore: ℒ~SDL(WD∗,WE∗)=O(M2(1−S)) L_SDL(W_D^*,W_E^*)=O(M^2(1-S)) (65) This completes the proof. ∎ I.4 Proof of Theorem 3.4 Proof. We prove both directions of the equivalence. Sufficient condition (⇒ ). Suppose wrd=WDσ(WEwpd)w_r^d=W_Dσ(W_Ew_p^d) for all d∈[n]d∈[n]. By definition of the approximate loss: ℒ~SDL(WD,WE) L_SDL(W_D,W_E) =∑d=1nMd‖wrd−WDσ(WEwpd)‖2 = _d=1^nM_d\|w_r^d-W_Dσ(W_Ew_p^d)\|^2 (66) =∑d=1nMd‖wrd−wrd‖2 = _d=1^nM_d\|w_r^d-w_r^d\|^2 (67) =0 =0 (68) Necessary condition (⇐ ). Suppose ℒ~SDL(WD,WE)=0 L_SDL(W_D,W_E)=0. Then: ∑d=1nMd‖wrd−WDσ(WEwpd)‖2=0 _d=1^nM_d\|w_r^d-W_Dσ(W_Ew_p^d)\|^2=0 (69) Since Md>0M_d>0 for all d∈[n]d∈[n] (by definition, Md=Pr⁡((s)=xd(s)ed)⋅[xd(s)2|xd(s)>0]M_d= (x(s)=x_d(s)e_d)·E[x_d(s)^2|x_d(s)>0] where both factors are positive under Assumptions 2.3 and 2.4): ‖wrd−WDσ(WEwpd)‖2=0∀d∈[n]\|w_r^d-W_Dσ(W_Ew_p^d)\|^2=0 ∀ d∈[n] (70) Therefore: wrd=WDσ(WEwpd)∀d∈[n]w_r^d=W_Dσ(W_Ew_p^d) ∀ d∈[n] (71) ∎ I.5 Proof of Theorem 3.7 Proof. We construct a configuration (WD∗,WE∗)(W_D^*,W_E^*) exhibiting the polysemanticity pattern and explicitly verify that both gradients vanish. Step 1: Given encoder WE∗W_E^* from realizability. Since the activation pattern =(ℱ1,…,ℱnq)P=(F_1,…,F_n_q) is realizable and forms a partition of [n][n], by Definition 3.6, there exists an encoder WE∗∈ℝnq×npW_E^* ^n_q× n_p such that: ∀i∈[nq],ℱi=d∈[n]:(σ(WE∗wpd))i>0∀ i∈[n_q], _i= \d∈[n]: (σ(W_E^*w_p^d) )_i>0 \ (72) Fix this WE∗W_E^* and define the activation vectors: zd:=σ(WE∗wpd)∈ℝnq,d=1,…,nz_d:=σ(W_E^*w_p^d) ^n_q, d=1,…,n (73) Since P forms a partition, for each feature d there exists a unique neuron i(d)i(d) such that d∈ℱi(d)d _i(d), and: zd=⟨wE∗,i(d),wpd⟩⋅ei(d)z_d= w_E^*,i(d),w_p^d · e_i(d) (74) That is, only neuron i(d)i(d) activates for feature d. Step 2: Construct optimal decoder WD∗W_D^*. For each neuron i∈[nq]i∈[n_q], we construct the decoder column WD∗,iW_D^*,i by minimizing the loss over features activating that neuron. Dead neurons: If ℱi=∅F_i= , set WD∗,i=0W_D^*,i=0. Active neurons: If ℱi≠∅F_i≠ , solve: WD∗,i=arg⁡minWDi∈ℝnr∑d∈ℱiMd‖wrd−WDi(zd)i‖2W_D^*,i= _W_D^i ^n_r _d _iM_d \|w_r^d-W_D^i(z_d)_i \|^2 (75) This is a least squares problem with solution: WD∗,i=∑d∈ℱiMd(zd)iwrd∑d∈ℱiMd(zd)i2W_D^*,i= _d _iM_d(z_d)_iw_r^d _d _iM_d(z_d)_i^2 (76) Step 3: Verify ∇WDℒ~SDL(WD∗,WE∗)=0 _W_D L_SDL(W_D^*,W_E^*)=0. The loss is ℒ~SDL(WD,WE)=∑d=1nMd‖wrd−WDzd‖2 L_SDL(W_D,W_E)= _d=1^nM_d\|w_r^d-W_Dz_d\|^2. The gradient with respect to column WDiW_D^i is: ∇WDiℒ~SDL=−2∑d=1nMd(wrd−WDzd)(zd)i _W_D^i L_SDL=-2 _d=1^nM_d(w_r^d-W_Dz_d)(z_d)_i (77) Since P is a partition, (zd)i≠0(z_d)_i≠ 0 only when i=i(d)i=i(d), i.e., when d∈ℱid _i. Therefore: ∇WDiℒ~SDL(WD∗,WE∗)=−2∑d∈ℱiMd(wrd−WD∗,i(zd)i)(zd)i _W_D^i L_SDL(W_D^*,W_E^*)=-2 _d _iM_d(w_r^d-W_D^*,i(z_d)_i)(z_d)_i (78) By construction (76), WD∗,iW_D^*,i satisfies the normal equation: ∑d∈ℱiMd(zd)i(wrd−WD∗,i(zd)i)=0 _d _iM_d(z_d)_i(w_r^d-W_D^*,i(z_d)_i)=0 (79) Therefore ∇WDiℒ~SDL(WD∗,WE∗)=0 _W_D^i L_SDL(W_D^*,W_E^*)=0 for all i∈[nq]i∈[n_q]. Step 4: Verify ∇WEℒ~SDL(WD∗,WE∗)=0 _W_E L_SDL(W_D^*,W_E^*)=0. Within the activation pattern region Ω _A, for feature d∈ℱid _i, the activation is: zd=⟨wEi,wpd⟩⋅eiz_d= w_E^i,w_p^d · e_i (80) The gradient with respect to encoder row wEiw_E^i is: ∇wEiℒ~SDL=∑d=1nMd∇wEi‖wrd−WDzd‖2 _w_E^i L_SDL= _d=1^nM_d _w_E^i \|w_r^d-W_Dz_d \|^2 (81) Only features d∈ℱid _i have non-zero contribution. For such d: ∂zd∂wEi=wpd⊗ei ∂ z_d∂ w_E^i=w_p^d e_i (82) where ⊗ denotes outer product. Therefore: ∇wEi‖wrd−WDzd‖2 _w_E^i \|w_r^d-W_Dz_d \|^2 =−2(wrd−WDzd)⊤WD∂zd∂wEi =-2(w_r^d-W_Dz_d) W_D ∂ z_d∂ w_E^i (83) =−2(wrd−WD∗,i⟨wE∗,i,wpd⟩)⊤WD∗,iwpd =-2(w_r^d-W_D^*,i w_E^*,i,w_p^d ) W_D^*,iw_p^d (84) The total gradient is: ∇wEiℒ~SDL _w_E^i L_SDL =−2∑d∈ℱiMd(wrd−WD∗,i⟨wE∗,i,wpd⟩)⊤WD∗,iwpd =-2 _d _iM_d(w_r^d-W_D^*,i w_E^*,i,w_p^d ) W_D^*,iw_p^d (85) =−2(WD∗,i)⊤∑d∈ℱiMd⟨wE∗,i,wpd⟩(wrd−WD∗,i⟨wE∗,i,wpd⟩) =-2(W_D^*,i) _d _iM_d w_E^*,i,w_p^d (w_r^d-W_D^*,i w_E^*,i,w_p^d ) (86) From the normal equation in Step 3 with (zd)i=⟨wE∗,i,wpd⟩(z_d)_i= w_E^*,i,w_p^d : ∑d∈ℱiMd⟨wE∗,i,wpd⟩(wrd−WD∗,i⟨wE∗,i,wpd⟩)=0 _d _iM_d w_E^*,i,w_p^d (w_r^d-W_D^*,i w_E^*,i,w_p^d )=0 (87) Therefore: ∇wEiℒ~SDL(WD∗,WE∗)=−2(WD∗,i)⊤⋅0=0 _w_E^i L_SDL(W_D^*,W_E^*)=-2(W_D^*,i) · 0=0 (88) This holds for all active neurons. For dead neurons (ℱi=∅F_i= ), the gradient is automatically zero. Step 5: (WD∗,WE∗)(W_D^*,W_E^*) is a partial optimum. By Theorem 3.2, ℒ~SDL L_SDL is convex in WDW_D for fixed WE∗W_E^*, and convex in WEW_E for fixed WD∗W_D^* within Ω _A. Since we have verified: • ∇WDℒ~SDL(WD∗,WE∗)=0 _W_D L_SDL(W_D^*,W_E^*)=0 (Step 3) • ∇WEℒ~SDL(WD∗,WE∗)=0 _W_E L_SDL(W_D^*,W_E^*)=0 within Ω _A (Step 4) By the first-order optimality condition for convex functions: • WD∗W_D^* minimizes ℒ~SDL(⋅,WE∗) L_SDL(·,W_E^*) • WE∗W_E^* minimizes ℒ~SDL(WD∗,⋅) L_SDL(W_D^*,·) over Ω _A By standard results in biconvex optimization (Gorski et al., 2007), (WD∗,WE∗)(W_D^*,W_E^*) is a partial optimum. Step 6: Non-zero loss. Since the pattern is polysemantic, there exists i∈[nq]i∈[n_q] with |ℱi|≥2|F_i|≥ 2. Choose distinct d1,d2∈ℱid_1,d_2 _i. By Theorem 3.4, zero loss requires: wrd1=WD∗zd1=WD∗,i(zd1)i,wrd2=WD∗zd2=WD∗,i(zd2)iw_r^d_1=W_D^*z_d_1=W_D^*,i(z_d_1)_i, w_r^d_2=W_D^*z_d_2=W_D^*,i(z_d_2)_i (89) If both equations held, then: wrd1(zd1)i=WD∗,i=wrd2(zd2)i w_r^d_1(z_d_1)_i=W_D^*,i= w_r^d_2(z_d_2)_i (90) Since (zd1)i=⟨wE∗,i,wpd1⟩>0(z_d_1)_i= w_E^*,i,w_p^d_1 >0 and (zd2)i=⟨wE∗,i,wpd2⟩>0(z_d_2)_i= w_E^*,i,w_p^d_2 >0, this requires: wrd1=(zd1)i(zd2)iwrd2w_r^d_1= (z_d_1)_i(z_d_2)_iw_r^d_2 (91) This means wrd1∝wrd2w_r^d_1 w_r^d_2, contradicting Assumption 2.4 (distinct features have linearly independent reconstruction vectors). Therefore: ℒ~SDL(WD∗,WE∗)=∑d=1nMd‖wrd−WD∗zd‖2>0 L_SDL(W_D^*,W_E^*)= _d=1^nM_d\|w_r^d-W_D^*z_d\|^2>0 (92) Conclusion. We have constructed (WD∗,WE∗)(W_D^*,W_E^*) exhibiting the polysemanticity pattern, explicitly verified both gradients vanish, and shown it achieves positive loss, completing the proof. ∎ I.6 Proof of Theorem 3.10 Proof. We construct a realizable pattern exhibiting feature absorption by scaling down the parent neuron’s encoder until one sub-concept separates, then adding a dedicated neuron for the separated feature. Step 1: Initial configuration and feature geometry. Since =(ℱ1,…,ℱM)P=(F_1,…,F_M) is realizable, there exists an encoder WE∈ℝM×npW_E ^M× n_p and threshold c>0c>0 such that: ∀i∈[M],ℱi=d∈[n]:⟨wEi,wpd⟩>c∀ i∈[M], _i=\d∈[n]: w_E^i,w_p^d >c\ (93) Fix any parent neuron i∗∈[M]i^*∈[M] with sub-concepts ℱi∗=di∗,1,…,di∗,ki∗F_i^*=\d_i^*,1,…,d_i^*,k_i^*\ where ki∗≥2k_i^*≥ 2. For each sub-concept, compute its activation strength: aj:=⟨wEi∗,wpdi∗,j⟩>c,∀j∈[ki∗]a_j:= w_E^i^*,w_p^d_i^*,j >c, ∀ j∈[k_i^*] (94) Since there are finitely many sub-concepts, define: amin:=minj∈[ki∗]⁡aj>ca_ := _j∈[k_i^*]a_j>c (95) Let j∗∈[ki∗]j^*∈[k_i^*] be the index achieving this minimum: aj∗=amina_j^*=a_ . Step 2: Scale down parent neuron to separate one sub-concept. We shrink the encoder row wEi∗w_E^i^* by multiplying with scaling factor λ∈(0,1)λ∈(0,1): w~Ei∗:=λ⋅wEi∗ w_E^i^*:=λ· w_E^i^* (96) The new activation strengths become: a~j:=⟨w~Ei∗,wpdi∗,j⟩=λ⋅aj a_j:= w_E^i^*,w_p^d_i^*,j =λ· a_j (97) Choose λ such that: λ:=c+amin2amin∈(camin,1)λ:= c+a_ 2a_ ∈ ( ca_ ,1 ) (98) This gives: • For the minimum-activation feature: a~j∗=λamin=c+amin2<c+ϵ a_j^*=λ a_ = c+a_ 2<c+ε for any small ϵ>0ε>0 • For all other features: a~j=λaj>λamin=c+amin2 a_j=λ a_j>λ a_ = c+a_ 2 By choosing the threshold to be c′=c+amin2c = c+a_ 2, we have: a~j∗=c′(at boundary),a~j>c′ for all j≠j∗ a_j^*=c (at boundary), a_j>c for all j≠ j^* (99) Slightly increasing c′c by infinitesimal ϵ>0ε>0 gives c′=c′+ϵc =c +ε, ensuring: a~j∗<c′<a~j∀j≠j∗ a_j^*<c < a_j ∀ j≠ j^* (100) Therefore, with encoder row w~Ei∗ w_E^i^* and threshold c′c : d∈[n]:⟨w~Ei∗,wpd⟩>c′=ℱi∗∖di∗,j∗\d∈[n]: w_E^i^*,w_p^d >c \=F_i^* \d_i^*,j^*\ (101) Step 3: Construct dedicated neuron for separated feature. We create a new encoder row parallel to the separated feature: wEnew:=α⋅wpdi∗,j∗w_E^new:=α· w_p^d_i^*,j^* (102) where α>0α>0 is chosen sufficiently large. By the unit-norm assumption (‖wpdi∗,j∗‖=1\|w_p^d_i^*,j^*\|=1): ⟨wEnew,wpdi∗,j∗⟩=α‖wpdi∗,j∗‖2=α w_E^new,w_p^d_i^*,j^* =α\|w_p^d_i^*,j^*\|^2=α (103) For any other feature d≠di∗,j∗d≠ d_i^*,j^*: ⟨wEnew,wpd⟩=α⟨wpdi∗,j∗,wpd⟩≤αM w_E^new,w_p^d =α w_p^d_i^*,j^*,w_p^d ≤α M (104) where M<1M<1 is the maximum interference from Assumption 2.4. Choose α large enough such that: α>c′,αM<c′α>c , α M<c (105) This is possible by taking α=2c′1+Mα= 2c 1+M, giving: α=2c′1+M>c′,αM=2c′M1+M<c′α= 2c 1+M>c , α M= 2c M1+M<c (106) where the first inequality uses M<1M<1 and the second uses M<1M<1. Therefore: • ⟨wEnew,wpdi∗,j∗⟩=α>c′ w_E^new,w_p^d_i^*,j^* =α>c (separated feature activates new neuron) • ⟨wEnew,wpd⟩≤αM<c′ w_E^new,w_p^d ≤α M<c for all d≠di∗,j∗d≠ d_i^*,j^* (no other features activate) Step 4: The new activation pattern. Construct the expanded encoder WE′∈ℝ(M+1)×npW_E ^(M+1)× n_p: WE′=[wE1⋮wEi∗−1w~Ei∗wEi∗+1⋮wEMwEnew]W_E = bmatrixw_E^1\\ \\ w_E^i^*-1\\ w_E^i^*\\ w_E^i^*+1\\ \\ w_E^M\\ w_E^new bmatrix (107) With threshold c′c , the resulting activation pattern is: ′=(ℱ1,…,ℱi∗−1,ℱi∗∖di∗,j∗,ℱi∗+1,…,ℱM,di∗,j∗)P =(F_1,…,F_i^*-1,F_i^* \d_i^*,j^*\,F_i^*+1,…,F_M,\d_i^*,j^*\) (108) This pattern is realizable by construction. ∎

Instant research discovery