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Equal Path Cost, Unequal Output Effects: Understanding Perturbation Propagation in Diffusion Models
Wei Guo, Yaowen Zhang, Xingtong Ge, Jun Zhang
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 92%
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Summary
This paper presents a theoretical framework to understand how perturbations in diffusion model sampling trajectories propagate to affect generated outputs. The authors define 'path cost' as the KL divergence between perturbed and reference trajectory distributions, showing it bounds but does not determine output changes. They derive a response identity linking local perturbations to feature mean responses, explaining why equal path costs can yield unequal output effects. The framework is validated through controlled interventions and applied to diagnose errors in cache-based acceleration methods.
Entities (8)
Relation Signals (5)
Path Cost → isdefinedby → Kullback-Leibler divergence
confidence 95% · quantify perturbation strength using the Kullback--Leibler (KL) divergence... termed as path cost
Path Cost → bounds → Output Distribution Changes
confidence 93% · path cost, which is shown to bound, but do not determine, changes in the output distribution.
Perturbation Propagation → determines → Feature Mean Response
confidence 91% · derive a response identity that connects the propagation and accumulation of local perturbations with the information captured by a selected feature mean
Theoretical Framework → appliesto → Cache-based Acceleration
confidence 88% · apply it to cache-based acceleration and show that our propagation analysis reliably identifies sampling intervals where caching causes larger image errors.
DeepCache → isexampleof → Cache-based Acceleration
confidence 85% · DeepCache reuses high-level features... Caching and sensitivity-based acceleration.
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Abstract
Abstract:Diffusion models have achieved remarkable success in generative modeling, with their sampling procedures routinely modified to control generation and improve efficiency. These modifications introduce perturbations along the sampling trajectory, raising a central question: how do such perturbations affect generated output? To address this question, we develop a theoretical framework to investigate perturbation propagation, combining dynamical analysis of the sampling process with an information-theoretic characterization of output responses. Within this framework, we quantify perturbation strength using the Kullback--Leibler (KL) divergence between perturbed and reference trajectory distributions, termed as path cost, which is shown to bound, but do not determine, changes in the output distribution. Building on this analysis, we derive a response identity that connects the propagation and accumulation of local perturbations with the information captured by a selected feature mean, explaining why changes in the output distribution can remain undetected by its first-order response. We test our theoretical analysis through controlled interventions at equal path cost in pretrained diffusion models, revealing distinct patterns of output sensitivity across sampling stages and spatial frequencies. To assess whether our framework can diagnose perturbations arising from practical approximations, we apply it to cache-based acceleration and show that our propagation analysis reliably identifies sampling intervals where caching causes larger image errors.
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- Source: https://arxiv.org/abs/2610.11380v1
- Canonical: https://arxiv.org/abs/2610.11380v1
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Equal Path Cost, Unequal Output Effects: Understanding Perturbation Propagation in Diffusion Models Wei Guo *Equal contribution. †Corresponding author. Affiliation: iComAI Lab, The Hong Kong University of Science and Technology Yaowen Zhang *Equal contribution. †Corresponding author. Affiliation: Department of Physics, The Hong Kong University of Science and Technology Xingtong Ge *Equal contribution. †Corresponding author. Affiliation: iComAI Lab, The Hong Kong University of Science and Technology Jun Zhang *Equal contribution. †Corresponding author. Affiliation: iComAI Lab, The Hong Kong University of Science and Technology Abstract Diffusion models have achieved remarkable success in generative modeling, with their sampling procedures routinely modified to control generation and improve efficiency. These modifications introduce perturbations along the sampling trajectory, raising a central question: how do such perturbations affect generated output? To address this question, we develop a theoretical framework to investigate perturbation propagation, combining dynamical analysis of the sampling process with an information-theoretic characterization of output responses. Within this framework, we quantify perturbation strength using the Kullback–Leibler (KL) divergence between perturbed and reference trajectory distributions, termed as path cost, which is shown to bound, but do not determine, changes in the output distribution. Building on this analysis, we derive a response identity that connects the propagation and accumulation of local perturbations with the information captured by a selected feature mean, explaining why changes in the output distribution can remain undetected by its first-order response. We test our theoretical analysis through controlled interventions at equal path cost in pretrained diffusion models, revealing distinct patterns of output sensitivity across sampling stages and spatial frequencies. To assess whether our framework can diagnose perturbations arising from practical approximations, we apply it to cache-based acceleration and show that our propagation analysis reliably identifies sampling intervals where caching causes larger image errors. 1 Introduction Diffusion models generate samples by simulating learned reverse-time dynamics from a noise distribution to the data distribution (Ho et al., 2020; Song et al., 2020). During inference, these dynamics are perturbed by errors in learned score estimates or denoising predictions (Song et al., 2021; Li and van der Schaar, 2024), by deliberate guidance updates (Dhariwal and Nichol, 2021), and by approximations introduced by fast solvers, quantization, and feature caching (Lu et al., 2022; Li et al., 2023; Ma et al., 2024). Because a perturbation changes the input to subsequent denoising steps, its effect on the generated sample depends on the downstream sampling dynamics, not only on its magnitude at the timestep where it occurs (Li and van der Schaar, 2024; Liu et al., 2026). This creates a mismatch between the trajectory-level quantities used to characterize a perturbation and the output-level quantities used to evaluate its consequences: the former describe the change where it enters the sampling process, whereas the latter concern the endpoint distribution, the response of a selected feature, or the displacement of a generated sample. This mismatch raises a central question: How do perturbations along the sampling trajectory translate into changes in generated outputs, and to what extent can trajectory-level measures characterize those changes? Existing work has addressed different aspects of this question. Analyses of score estimation and discretization use aggregated errors along the sampling trajectory to bound deviations in the generated distribution (Song et al., 2021; Chen et al., 2022; Benton et al., 2024). Studies of exposure bias and error propagation instead examine how prediction errors accumulate across denoising steps (Ning et al., 2023; Ning et al., 2024; Li and van der Schaar, 2024). More recently, in the setting of recursively trained diffusion models, Khelifa et al. (2026) derive terminal-divergence lower bounds parameterized by an endpoint-observability coefficient, which measures how much pathwise score-error energy remains detectable in the terminal distribution. These results show that inaccuracies along the sampling trajectory can accumulate into the global generation error and the pathwise error may be only partially observable in the terminal distribution. However, they do not fully characterize how the downstream sampling dynamics mediate the relationship between the magnitude of a perturbation along the trajectory and its effect on the final generated output. To address this gap, we develop a theoretic framework combining information-theoretic analysis with a dynamical characterization of output response, which explains why perturbations of equal strength can produce unequal output effects and derives how local perturbations propagate and accumulate to determine output responses. We measure perturbation strength by the KL divergence between perturbed and reference trajectory distributions, which we call path cost. The analysis shows that a perturbation’s effects on generated outputs depend jointly on its path cost, how it propagates and interacts with perturbations at other steps, and how the resulting changes are measured. Our contributions are: • A theoretical framework explaining how perturbations affect generated outputs. We derive an exact decomposition of path cost that distinguishes changes in the output distribution from changes in trajectory distributions conditional on the output. We derive a first-order propagation law that quantifies how subsequent sampling dynamics transform local perturbations and how their contributions accumulate to determine the final output response. We further establish bounds relating feature mean responses to local output-distribution changes, explaining why a selected feature mean can fail to detect such changes. These results explain why equal path costs can produce unequal output effects. • Experimental support for the theoretical analysis. We test the framework through controlled perturbations at equal path cost, varying the sampling stages at which they are applied and their spatial directions. The resulting output responses differ across sampling stages and directions. The experiments show that first-order propagation can capture the temporal patterns of responses to finite perturbations, although predicted and observed magnitudes can differ substantially. They also show that smaller absolute feature mean responses can correspond to larger changes relative to baseline fluctuations, consistent with our theoretical analysis. • Applying the framework to diagnose output errors in cache-based acceleration. We apply the framework to cache-based acceleration for diffusion generation, where earlier model predictions are reused to avoid fresh evaluations. By tracing how the resulting approximation errors propagate and accumulate, we assess their impact on the final image and identify sampling intervals where caching causes larger output deviations. This propagation-based diagnostic identifies such intervals more reliably than local error magnitude and path-cost measures, which remains informative even when predicted and observed error magnitudes differ substantially. 2 Related work Error propagation in diffusion sampling. Path-space control relates diffusion modeling to KL minimization (Berner et al., 2024), while score-error analyses provide likelihood and sampling-error bounds (Song et al., 2021; Chen et al., 2022; Benton et al., 2024). Li and van der Schaar (2024) relate local and cumulative errors through a propagation equation. Khelifa et al. (2026) quantify the endpoint visibility of pathwise score errors to derive terminal-divergence lower bounds, with applications to recursive training. We study perturbations of a fixed sampler, connecting output visibility to downstream propagation and the information captured by a selected feature mean. Conditional information and observable response. Conditional-score projections characterize Fisher information retained by an observation (Pollard, 2013), while fluctuation–response bounds relate mean sensitivity to variance and path information (Dupuis et al., 2016; Dechant and Sasa, 2020). De Bortoli et al. (2025) bound diffusion sampling errors using smooth test functions. We build on these information and response identities to characterize how much path information reaches the output and how much a feature mean captures, with propagation determining its response. Spectral generation dynamics. Benita et al. (2025) derive Gaussian spectral transfer dynamics and use data spectra to design schedules. We compare temporal and directional responses at matched path cost, and distinguish absolute response from response relative to baseline feature fluctuations. Caching and sensitivity-based acceleration. DeepCache reuses high-level features (Ma et al., 2024); SenCache uses denoiser sensitivity to latent and timestep changes (Haghighi and Alahi, 2026). ERTACache analyzes and corrects feature-shift and step-amplification errors (Peng et al., 2026), while EpaCache calibrates reuse thresholds from terminal deviations under single-step caching interventions (Liu et al., 2026). We use reference-trajectory propagation to diagnose image errors from multi-step output caching and test whether it identifies sampling windows with larger errors more reliably than local-defect and path-cost measures. 3 A Theoretical Framework for Perturbation Propagation In this section, we derive a theoretical characterization of how local perturbations propagate through diffusion sampling and affect measured output responses. We first quantify trajectory changes by path cost, separate output changes from conditional trajectory changes, and derive perturbation propagation. Our main theorem expresses the measured response through the propagated local perturbations and identifies the fractions of path information retained in the output distribution and captured by the chosen feature mean. 3.1 How path cost quantifies perturbation strength Diffusion models learn to reverse a forward process that progressively adds noise to data. Generation starts from noise and follows learned reverse dynamics through discrete sampling updates (Ho et al., 2020; Song et al., 2020). Common stochastic diffusion samplers update the state using a model-dependent mean and Gaussian noise (Nichol and Dhariwal, 2021). These updates can then be expressed in the following form Xi+1=i0(Xi,ξi)=μi(Xi)+Li(Xi)ξi,ξi∼i.i.d.(0,I).X_i+1=G_i^0(X_i, _i)= _i(X_i)+L_i(X_i) _i, _i i.i.d. N(0,I). (1) Here, XiX_i denotes the sampling state after i generation steps, with X0X_0 drawn from the initial noise distribution and XMX_M denoting the terminal sampling state. The index i=0,…,M−1i=0,…,M-1 follows generation order from noise toward the output. The map i0G_i^0 defines one update of the reference sampler with the superscript 00 indicating the unperturbed process.The variables ξi _i are independent standard Gaussian noise vectors, and Li(Xi)ξiL_i(X_i) _i is the stochastic component of the sampling update, with conditional covariance Λi(Xi)=Li(Xi)Li(Xi)⊤ _i(X_i)=L_i(X_i)L_i(X_i) . As mentioned above, local perturbations, such as prediction errors and computational approximations, modify the sampling updates in Eq. (1). We model the resulting dynamics by a family of perturbed sampling processes parameterized by perturbation strength ε , with ε=0 =0 corresponding to the reference process. To study how local perturbations affect the sampling process, we consider the complete trajectories Γ=(X0,…,XM) =(X_0,…,X_M) and denote their joint distribution under the reference and perturbed processes as P and QεQ_ , respectively, so that Q0=PQ_0=P. Such effects can be measured through the statistical distinguishability of these trajectory distributions using their Kullback–Leibler (KL) divergence (Cover and Thomas, 2006), which we define as the path cost, (ε)=DKL(Qε∥P)C( )=D_KL(Q_ \|P). We consider perturbations that preserve the initial distribution and shift each Gaussian transition mean by εui u_i, while retaining its positive definite covariance Λi _i. The resulting transition densities are strictly positive, ensuring that QεQ_ is absolutely continuous with respect to P, i.e., Qε≪PQ_ P (Kallenberg, 1997). Then, by using the relative-entropy chain rule (Cover and Thomas, 2006), we can obtain the expression of the path cost as a sum of conditional transition KLs as (ε):=DKL(Qε∥P)=ε22Qε∑iui⊤Λi−1ui.C( ):=D_KL(Q_ \|P)= ^22E_Q_ _iu_i _i^-1u_i. (2) This expression shows how local mean perturbations contribute to the change in the trajectory distribution. Each contribution is weighted by the inverse transition covariance: for Λi=νiI _i= _iI, the conditional KL at step i is ε2‖ui‖2/(2νi) ^2\|u_i\|^2/(2 _i). Thus, the same mean shift produces a larger change in the transition distribution, as measured by KL divergence, when the sampling noise is lower. The derivation details are provided in Appendix A.1. 3.2 Why path cost does not determine output changes Figure 1: Equal path costs, different output-distribution changes. Two temporal perturbation sequences act only on the first coordinate of a two-dimensional Gaussian reverse diffusion, each with path cost =2C=2 nats. Their contributions reinforce (top) or cancel (bottom) at the output. (a) Coordinate density evolution on a shared color scale, with perturbed (white) and reference (dashed) means. (b) Perturbed (solid) and reference (dashed) output densities. (c) Output KL (colored) and bridge cost (gray). Construction details are in Appendix B. Having quantified how perturbations change the sampling trajectory distribution, we now examine their effect on the distribution of generated outputs. The terminal sampling state XMX_M may require a final deterministic transformation, such as decoding a latent representation, to produce the generated output. We therefore write Y=ϕ(XM)Y=φ(X_M), where ϕφ denotes this transformation and is the identity when XMX_M is already the output. We use the same map for both the reference and perturbed processes. Then, we derive an exact decomposition of path cost into the change in the output distribution and the change in trajectory distributions conditional on the output in the following Proposition 1. Proposition 1 (Decomposition of path and output changes). For Qε≪PQ_ P with finite path KL and regular conditional path distributions, let pYp_Y and qε,Yq_ ,Y be the output distributions under P and QεQ_ . Then, the path cost can be decomposed as (ε)=DKL(qε,Y∥pY)⏟Y(ε)+qε,YDKL(Qε(⋅∣Y)∥P(⋅∣Y))⏟bridge|Y(ε).C( )= D_KL(q_ ,Y\|p_Y)_C_Y( )+ E_q_ ,YD_KL\! (Q_ (· Y)\|P(· Y) )_C_bridge Y( ). (3) Proof. Please see Appendix A.2. ∎ The above decomposition separates two consequences of a perturbation. The first term, Y(ε)C_Y( ), measures the change distinguishable from generated outputs. The second measures how the trajectory distributions differ when conditioned on the same output, averaged over the perturbed output distribution, which we call as the bridge cost. Since bridge cost is non-negative, we have 0≤Y(ε)≤(ε)0 _Y( ) ( ). Therefore, the path cost bounds the changes in output distribution, but equal path costs can yield different output changes depending on their bridge components. To illustrate this decomposition, Figure 1 presents a Gaussian toy example that we construct to show how perturbations with equal path costs can produce different output effects. In this example, scaling the cancelling perturbation sequence yields any positive path cost while output KL remains zero. Thus, a positive path cost alone does not imply a positive output-distribution change. To understand how perturbations from different steps reinforce or cancel at the output, we next examine their propagation through the sampling dynamics. 3.3 How local perturbations propagate and accumulate To trace how local perturbations affect generated outputs, we analyze the first-order response with respect to perturbation strength at ε=0 =0. This gives a linear description of propagation along the reference trajectories. Let Xi+1ε=iε(Xiε,ξi)X_i+1 =G_i (X_i , _i) denote the perturbed update, with iε|ε=0=i0G_i |_ =0=G_i^0 as in Eq. (1). To compare corresponding trajectories, we hold the initial state and noise variables fixed. Along each reference trajectory, we define Ji=Dxi0J_i=D_xG_i^0 as the Jacobian matrix of the reference update with respect to its input state, which describes how an input-state change propagates to the next step. We also define ei=∂εiε|0e_i= . _ G_i |_0, as the derivative of the sampling update with respect to perturbation strength with its input state and noise remained fixed. This term measures the local perturbation introduced by the update itself. For the mean perturbations in Section 3.1, ei=uie_i=u_i. To examine how local perturbations affect a particular property of the generated output, we represent that property by a scalar feature f(Y)f(Y), such as a spatial-frequency component of an image(Ahmed et al., 1974). Its expectation summarizes this property across generated samples. Therefore, we study how the expected feature value changes under perturbation and define the feature mean response (Kubo et al., 2012) as χf=∂εQεf(Y)|0 _f= . _ E_Q_ f(Y) |_0. Using pathwise differentiation (Mohamed et al., 2020), we express both the output response and the feature mean response in the following proposition. Proposition 2 (Perturbation propagation dynamics). Assume that iεG_i is differentiable in the state and perturbation strength, and that ϕφ is differentiable at the terminal state, almost surely along reference trajectories. The output response, defined as the derivative of the generated output with respect to perturbation strength at ε=0 =0, satisfies V:=∂εYε|0=∑i=0M−1Tiei,Ti=Dϕ(XM)JM−1⋯Ji+1,V:= . _ Y |_0= _i=0^M-1T_ie_i, T_i=Dφ(X_M)J_M-1·s J_i+1, (4) where TiT_i propagates the local perturbation at step i through the remaining sampling updates and the output map ϕφ. Furthermore, if f is continuously differentiable and [f(Yε)−f(Y)]/ε[f(Y )-f(Y)]/ is dominated by an integrable random variable near ε=0 =0, then the feature mean response is given as χf=P[∇f(Y)⊤V]=∑i=0M−1P[∇f(Y)⊤Tiei]. _f=E_P[∇ f(Y) V]= _i=0^M-1E_P[∇ f(Y) T_ie_i]. (5) Proof. Please see Appendix A.3. ∎ Proposition 2 shows how downstream dynamics transform each local perturbation through TiT_i, changing its strength and direction at the output. The propagated perturbations from different steps add to form the total output response V. To examine how their magnitudes and relative directions determine the squared magnitude of this response, we write vi=Tieiv_i=T_ie_i and expand the norm of V as ‖V‖2=∑i‖vi‖2+2∑i<j⟨vi,vj⟩.\|V\|^2= _i\|v_i\|^2+2 _i<j v_i,v_j . (6) The first term sums the squared magnitudes of the propagated perturbations considered separately. The cross terms account for their relative directions: positive inner products increase the total squared response, while negative inner products reduce it. Thus, the magnitude of the total response depends on both the sizes of the propagated perturbations and how their directions align at the output. The above analysis explains how local perturbations propagate and accumulate to determine the output response and the mean response of a selected feature. However, this feature mean describes only one property of the output distribution. Interpreting its response therefore requires understanding how much distributional change it captures and how this depends on the choice of feature. 3.4 What feature mean responses reveal about output changes To identify the distributional changes captured by the feature mean, we derive its response in terms of trajectory likelihoods. Differentiating Qεf(Y)=P[f(Y)dQε/dP]E_Q_ f(Y)=E_P[f(Y)\,dQ_ /dP] with respect to perturbation strength at ε=0 =0, we obtain χf=P[f(Y)S], _f=E_P[f(Y)S], (7) where S=∂εlog(dQε/dP)|0S= . _ (dQ_ /dP) |_0 is the path likelihood score, describing changes in trajectory log likelihood with perturbation strength. To identify the leading dependence of path cost on perturbation strength, we derive the expansion of Eq. (2) near the reference sampler as (ε)=12ε2Fpath+o(ε2),C( )= 12 ^2F_path+o( ^2), (8) where Fpath=PS2F_path=E_PS^2 is the path Fisher information (Amari, 2016; Dupuis et al., 2016). Since f(Y)f(Y) depends only on the output, we can average S over trajectories sharing that output without changing χf _f. This gives the output likelihood score SY=P[S∣Y]S_Y=E_P[S Y] (Pollard, 2013). We define the output Fisher information as FY=PSY2F_Y=E_PS_Y^2 and derive the second-order expansion of the KL decomposition in Eq. (3), yielding Fpath=FY+PVarP(S∣Y)⏟Fbridge|Y.F_path=F_Y+ E_PVar_P(S Y)_F_bridge Y. (9) The first term determines local output-distribution change through Y(ε)=12ε2FY+o(ε2)C_Y( )= 12 ^2F_Y+o( ^2), while the second is the local bridge component. For Fpath>0F_path>0, the ratio FY/FpathF_Y/F_path therefore gives the fraction of path cost visible at the output as ε→0 → 0. The assumptions and derivations for Eqs. (8) and (9) are provided in Appendix A.4. To compare the feature mean response with the output and path Fisher information, we define the squared normalized mean response as Bf=χf2/VarP(f(Y))B_f= _f^2/Var_P(f(Y)). For small ε , the squared mean shift on this scale is ε2Bf+o(ε2) ^2B_f+o( ^2), matching the perturbation order of the local KL expansions. We derive a lower bound on output Fisher information from this normalized response using the fluctuation–response bound (Dupuis et al., 2016; Dechant and Sasa, 2020) in the following Proposition 3. Proposition 3 (Feature response and output information). Assume that the path likelihood ratio is differentiable in mean square at ε=0 =0, meaning P[(dQε/dP−1−εS)2]=o(ε2)E_P[(dQ_ /dP-1- S)^2]=o( ^2), with PS2<∞E_PS^2<∞. Then any real-valued output feature f(Y)f(Y) with 0<VarP(f(Y))<∞0<Var_P(f(Y))<∞ satisfies χf=P[f(Y)S]=P[f(Y)SY],Bf=χf2VarP(f(Y))≤FY≤Fpath. _f=E_P[f(Y)S]=E_P[f(Y)S_Y], B_f= _f^2Var_P(f(Y))≤ F_Y≤ F_path. (10) Proof. Please see Appendix A.5. ∎ Proposition 3 identifies two stages at which perturbation information can be lost, from the trajectory to the output distribution and from the output distribution to the selected feature mean. We derive an exact expression for the fraction of path information captured by the chosen feature mean, incorporating both perturbation propagation and the two stages of information loss in the following Theorem 1. Theorem 1 (Perturbation propagation and measured output response). Under the assumptions of Propositions 2 and 3, let f be continuously differentiable with 0<VarP(f(Y))<∞0<Var_P(f(Y))<∞. If FY>0F_Y>0, the fraction of path information captured by the feature mean response satisfies BfFpath=(∑iP[∇f(Y)⊤Tiei])2VarP(f(Y))Fpath=FYFpath⏟output visibilityCorrP(f(Y),SY)2⏟feature–score alignment. B_fF_path= ( _iE_P[∇ f(Y) T_ie_i] )^2Var_P(f(Y))\,F_path= F_YF_path_output visibility Corr_P(f(Y),S_Y)^2_feature--score alignment. (11) Proof. Please see Appendix A.6. ∎ Theorem 1 explains why perturbations with equal path cost can produce different measured responses. The propagated terms show how downstream dynamics, cross-time accumulation, and the selected feature jointly shape the response. Here, output visibility measures the fraction of path cost reflected in output-distribution change for small perturbations, and feature–score alignment measures the fraction of local output information captured by the chosen feature mean. In particular, a perturbation can change the output distribution while leaving the mean of the chosen feature unchanged to first order in perturbation strength. 4 Controlled Perturbations Propagation with Equal Path Cost In this section, we test the theoretical framework by using controlled perturbations at equal path cost, which provides a common measure of perturbation strength for comparison. 4.1 Experimental setup Figure 2: Output responses at equal path cost within each model. Rows show perturbation frequencies and columns show the sampling windows in which perturbations are applied, from early to late. Each row is divided by its maximum response. White dots mark the peak windows and numbers on the right give the maxima before normalization. We study controlled perturbations in class-conditional ImageNet-256 generation using pixel-space JiT-B/L (Li and He, 2026) and ADM (Dhariwal and Nichol, 2021), and latent-space DiT (Peebles and Xie, 2023), without additional guidance. We group the 128 stochastic sampling steps into eight consecutive windows of 16 steps, numbered 1–8 in generation order from noise toward the output. In each experiment, we perturb the transition means within one window. To vary the spatial direction, we use unit-norm discrete cosine transform (DCT) templates, which represent spatial cosine patterns on the pixel or latent grid: DC is constant, and DCT-k has higher frequency as k increases. We normalize each perturbation to satisfy ∑iui⊤Λi−1ui=1 _iu_i _i^-1u_i=1, giving equal path cost (ε)=ε2/2C( )= ^2/2 across windows and directions within each model, conditional on class and template. We use ε=0.25 =0.25 for JiT/ADM and 0.80.8 for DiT. For JiT and ADM, we measure changes in the mean DCT coefficients of generated images. For DiT, we measure the RMS image response after decoding. Measurement details are provided in Appendix C. 4.2 Spatiotemporal sensitivity to perturbations Figure 2 shows how output responses vary with perturbation timing and spatial frequency at equal path cost. In JiT-B, all four frequencies shown produce their largest response in the first window, with responses decreasing toward later windows. In ADM, the peak shifts from window 1 for DC to window 4 for DCT-16; in DiT, it shifts from window 3 for DC to window 5 for DCT-31. Thus, earlier perturbations do not always produce larger output responses: the most sensitive stage depends on both the model and perturbation frequency. These observations are consistent with Section 3.3, where downstream propagation shapes the output effect of a perturbation beyond its path cost. We provide two DiT image examples in Appendix D.1, illustrating how perturbations with equal path cost produce different output changes when applied at different sampling stages. 4.3 Local predictions and finite perturbation responses Figure 3: Local predictions and finite perturbation responses in DiT. Mean decoded-image RMS of local derivatives and symmetric finite differences (ε=0.8 =0.8) on the same 32 trajectories, where shaded bands indicate one standard error of the mean across samples. The right panel reports finite/local peak ratios with dashed line denoting equal amplitudes. All runs use 128 stochastic steps. The responses above are measured at finite perturbation strength, whereas the propagation analysis in Section 3.3 describes the output response to infinitesimal perturbations. We use DiT to test how well this local prediction captures the magnitude and temporal pattern of responses to finite perturbations. On the same 32 sampling trajectories, we compare the mean per-sample RMS of the output derivative V, computed through the sampler and decoder using Jacobian–vector products (JVPs), with that of the symmetric finite difference (Y+ε−Y−ε)/(2ε)(Y^+ -Y^- )/(2 ) at ε=0.8 =0.8. As shown in Figure 3, the local and finite responses follow similar trends across sampling windows and peak at windows 3, 4, and 5 for DC, DCT-4, and DCT-31, respectively. However, the finite-response peaks are only 36.8%, 47.7%, and 87.9% of the corresponding local predictions, as shown in the right panel. Thus, in this DiT example, local propagation identifies the most responsive sampling windows, while overestimating the peak response magnitudes at the tested finite amplitude. 4.4 Feature responses relative to baseline fluctuations Having examined propagation through the decoded-image response in DiT, we now return to the feature mean responses in JiT and ADM to compare them on their baseline fluctuation scales. Section 3.4 relates a feature’s mean response to output information after normalization by its baseline variance. Table 1 compares absolute responses, baseline fluctuations, and normalized responses at the first sampling window. In JiT-B/L, DCT-16 has a smaller absolute response than DC, but its baseline fluctuations are sufficiently smaller to give a larger normalized response. In ADM, the absolute response decreases more sharply than the baseline fluctuations, so DCT-16 also has a smaller normalized response. These results show that a smaller mean response can represent a larger change relative to a feature’s baseline fluctuations, consistent with the analysis in Section 3.4. We provide the results for all sampling windows in Appendix D.2. Table 1: Feature responses relative to baseline fluctuations. Perturbations act at window 1 at equal path cost, with ε=0.25 =0.25 and N=512N=512 for JiT-B/L and 256256 for ADM. We report absolute mean response |χ^f|| χ_f|, baseline standard deviation sfs_f, and normalized response B^f=χ^f2/sf2 B_f= χ_f^2/s_f^2. JiT-B JiT-L ADM Feature |χ^f|| χ_f| sfs_f B^f B_f |χ^f|| χ_f| sfs_f B^f B_f |χ^f|| χ_f| sfs_f B^f B_f DC 3.313.31 127127 0.0006770.000677 3.363.36 114114 0.0008750.000875 70.770.7 106106 0.4470.447 DCT-1 2.412.41 48.548.5 0.002480.00248 2.462.46 45.245.2 0.002960.00296 24.924.9 42.542.5 0.3430.343 DCT-4 1.431.43 18.918.9 0.005770.00577 1.461.46 21.421.4 0.004650.00465 6.96.9 23.523.5 0.08650.0865 DCT-16 0.3840.384 3.883.88 0.009820.00982 0.3660.366 4.344.34 0.007120.00712 0.2740.274 4.324.32 0.004020.00402 5 Diagnosing Output Errors in Cache-Based Acceleration In this section, we investigate whether the perturbation propagation analysis in Section 3 can predict output errors caused by practical computational approximations during diffusion sampling. We consider cache-based acceleration as a practical example during diffusion sampling, since reusing earlier model predictions reduces the number of model evaluations but introduces approximation errors into subsequent updates (Ma et al., 2024; Liu et al., 2026). We divide 64 sampling steps into eight windows. Each candidate caches the prediction at the first step of one window B and reuses it for the remaining seven steps, while other steps use fresh predictions. The differences between cached and fresh predictions induce local update perturbations eie_i along the reference trajectory, whose detailed derivation is provided in Appendix C.4. Using their first-order propagation, we construct a predictor of the resulting image mean squared error (MSE) and compare its predictions with observed errors: Rpred=1d‖∑i∈BTiei‖2,Robs=‖Ycache−Y‖2d,R_pred= 1d \| _i∈ BT_ie_i \|^2, R_obs= \|Y^cache-Y\|^2d, (12) where d is the number of image entries and YcacheY^cache is the final output when predictions are cached within B. Figure 4: Predicted and observed image MSE under caching. Black solid lines show observed image MSE and blue dashed lines show predictions from the propagation-based predictor, averaged over the same 64 samples per model. Each candidate reuses the prediction cached at the start of one eight-step window, and windows are ordered from noise toward output. Table 2: Spearman correlations with observed mean image MSE across eight caching windows, using the runs in Figure 4. Bold values mark the highest correlation for each model. Model Our predictor Local defect Path cost JiT-B 0.9524 0.2143 -0.4762 JiT-L 0.9762 0.2381 -0.4286 ADM 0.9762 -0.9048 -0.9048 DiT 0.9762 0.5000 -0.3095 Figure 4 compares predicted and observed image MSE. The propagation-based predictor captures the broad variation in error across windows, although predicted and observed magnitudes can differ substantially. For JiT-B/L, the predictions also track the observed error magnitudes more closely, particularly in the middle and later windows. In Table 2, our propagation-based predictor achieves Spearman correlations of 0.95240.9524–0.97620.9762 with observed mean image MSE, substantially exceeding the local-defect and path-cost baselines defined in Appendix C.5 across all four models. 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We then prove the propagation identity, establish the local Fisher decomposition, and derive the feature-response bound and the main response identity. Throughout, P and QεQ_ denote the joint distributions of reference and perturbed sampling trajectories, respectively. A.1 Derivation of the path cost for local perturbations We derive how local changes in the sampling updates determine the distinguishability of complete trajectories. Starting from the Gaussian sampling rule in Eq. (1), we consider perturbations that shift the conditional transition mean while preserving the covariance function and initial distribution. We first express the trajectory likelihood ratio through the sampling transitions, then evaluate each transition’s contribution to obtain Eq. (2). Reference and perturbed transitions. We write Hi=(X0,…,Xi)H_i=(X_0,…,X_i) for the sampling history available before update i. Common conditioning variables, when present, are included in X0X_0 and remain fixed during generation. To allow a local perturbation to depend on previous sampling states, we write its mean displacement as εui(Hi) u_i(H_i). The reference and perturbed conditional transition distributions are Ki0(⋅∣Hi) K_i^0(· H_i) =(μi(Xi),Λi(Xi)), =N\! ( _i(X_i), _i(X_i) ), (13) Kiε(⋅∣Hi) K_i (· H_i) =(μi(Xi)+εui(Hi),Λi(Xi)). =N\! ( _i(X_i)+ u_i(H_i), _i(X_i) ). Here ui=0u_i=0 at unperturbed steps. We assume Λi(Xi) _i(X_i) is positive definite, so both conditional densities are strictly positive. Together with the shared initial distribution, this ensures Qε≪PQ_ P [Kallenberg, 1997]. We take the perturbations to have finite expected quadratic cost under QεQ_ , so the path KL derived below is finite. From trajectory likelihoods to transition costs. Let ki0k_i^0 and kiεk_i denote the densities of the two kernels in Eq. (13). Recall that Γ=(X0,…,XM) =(X_0,…,X_M) denotes a complete sampling trajectory. We factor the trajectory likelihood ratio into conditional density ratios: Lε(Γ):=dQεdP(Γ)=∏i=0M−1ℓiε(Xi+1,Hi),ℓiε(x,Hi):=kiε(x∣Hi)ki0(x∣Hi).L_ ( ):= dQ_ dP( )= _i=0^M-1 _i (X_i+1,H_i), _i (x,H_i):= k_i (x H_i)k_i^0(x H_i). (14) The common initial distribution cancels from this ratio. We take logarithms, average under QεQ_ , and condition each term on HiH_i to obtain (ε) ( ) =QεlogLε(Γ) =E_Q_ L_ ( ) (15) =∑i=0M−1Qε[Qε[logℓiε(Xi+1,Hi)∣Hi]] = _i=0^M-1E_Q_ [E_Q_ [ _i (X_i+1,H_i) H_i ] ] =∑i=0M−1QεDKL(Kiε(⋅∣Hi)∥Ki0(⋅∣Hi)). = _i=0^M-1E_Q_ D_KL\! (K_i (· H_i) \|K_i^0(· H_i) ). This follows from the relative-entropy chain rule [Cover and Thomas, 2006]. Each conditional KL compares the two possible updates at the same history; the outer expectation averages over histories reached by the perturbed process. Evaluating the Gaussian contribution. We now compute each conditional KL explicitly. At a fixed history, we abbreviate μi=μi(Xi) _i= _i(X_i), Λi=Λi(Xi) _i= _i(X_i), and ui=ui(Hi)u_i=u_i(H_i), and define the residual ri=Xi+1−μir_i=X_i+1- _i. Since the two Gaussian kernels share their covariance, their density normalization factors cancel. Expanding their quadratic exponents gives logℓiε(Xi+1,Hi) _i (X_i+1,H_i) =−12(ri−εui)⊤Λi−1(ri−εui)+12ri⊤Λi−1ri =- 12(r_i- u_i) _i^-1(r_i- u_i)+ 12r_i _i^-1r_i (16) =εui⊤Λi−1ri−ε22ui⊤Λi−1ui. = u_i _i^-1r_i- ^22u_i _i^-1u_i. Under the perturbed transition, Qε[ri∣Hi]=εuiE_Q_ [r_i H_i]= u_i. Taking the conditional expectation therefore yields DKL(Kiε(⋅∣Hi)∥Ki0(⋅∣Hi))=ε22ui⊤Λi−1ui.D_KL\! (K_i (· H_i) \|K_i^0(· H_i) )= ^22u_i _i^-1u_i. (17) Substituting into Eq. (15), we obtain the exact expression stated in Eq. (2): (ε)=DKL(Qε∥P)=ε22Qε∑i=0M−1ui⊤Λi−1ui.C( )=D_KL(Q_ \|P)= ^22E_Q_ _i=0^M-1u_i _i^-1u_i. (18) Thus the trajectory-level change is the expected sum of local mean displacements weighted by inverse transition covariance. For Λi=νiI _i= _iI, each contribution is ε2‖ui‖2/(2νi) ^2\|u_i\|^2/(2 _i), so an equal displacement has greater cost at a lower noise level. This identity holds at finite perturbation strength. The expectation remains under QεQ_ as the perturbation also changes which histories the sampler visits. A.2 Proof of Proposition 1: Decomposition of Path and Output Changes We prove Proposition 1 by factoring the trajectory likelihood ratio into output and conditional-trajectory likelihood ratios [Léonard, 2013]. We assume that the trajectory space is standard Borel and that Y is a measurable observation. Suppose Qε≪PQ_ P has finite KL, with likelihood Lε=dQε/dPL_ =dQ_ /dP. Under these assumptions, regular conditional laws exist. For every bounded measurable g, we have qε,Yg(Y)=P[Lεg(Y)]=P[P(Lε∣Y)g(Y)].E_q_ ,Yg(Y)=E_P[L_ g(Y)]=E_P[E_P(L_ Y)g(Y)]. (19) Identifying the output likelihood ratio, we obtain ℓε(Y):=dqε,YdpY(Y)=P[Lε∣Y]. _ (Y):= dq_ ,Ydp_Y(Y)=E_P[L_ Y]. (20) Thus, when only the output is observed, the path likelihood is averaged over all histories compatible with that output. Disintegrating QεQ_ and P with respect to Y gives dQε(⋅∣Y)dP(⋅∣Y)=Lεℓε(Y)Qε-a.s. dQ_ (· Y)dP(· Y)= L_ _ (Y) Q_ -a.s. (21) This proves Eq. (3): the first expectation is output KL, and the second is conditional-history KL averaged over the perturbed output law. Nonnegativity of both terms gives 0≤Y≤0 _Y . A.3 Proof of Proposition 2: Perturbation propagation We prove Proposition 2 by differentiating the discrete sampler on its actual sampling grid. Consider Xi+1ε=iε(Xiε,ξi)X_i+1 =G_i (X_i , _i) with the initial state and innovations held fixed. Assume the step maps are differentiable in state and perturbation strength along the reference trajectories, and ϕφ is differentiable at their endpoints, almost surely. For mean response, we additionally impose the integrability and interchange conditions in Appendix A.5. Piecewise differentiable maps are treated at their differentiable points. Forward transport. At fixed reference state and perturbation strength, we define Ji=Dxi0J_i=D_xG_i^0 and ei=∂εiε|0e_i= . _ G_i |_0. Define δXi=∂εXiε|0δ X_i= . _ X_i |_0. Applying the chain rule gives δXi+1=JiδXi+ei,δX0=0.δ X_i+1=J_iδ X_i+e_i, δ X_0=0. (22) The first term propagates state changes from earlier steps, while the second accounts for the local change in the current update. When ei=0e_i=0, an earlier state change can still propagate through JiJ_i. Iterating gives δXM=∑iJM−1⋯Ji+1ei.δ X_M= _iJ_M-1·s J_i+1e_i. (23) Differentiating Yε=ϕ(XMε)Y =φ(X_M ) then gives Eq. (4). For a scalar output feature, the first-order expansion of its mean is Qεf(Y)=Pf(Y)+εχf+o(ε),χf=∂εQεf(Y)|0.E_Q_ f(Y)=E_Pf(Y)+ _f+o( ), _f= . _ E_Q_ f(Y) |_0. (24) Under the stated interchange conditions, substituting the expression for V into χf=P[∇f(Y)⊤V] _f=E_P[∇ f(Y) V] yields Eq. (5). For mean controls, iε=i0+εuiG_i =G_i^0+ u_i and ei=ui(Xi)e_i=u_i(X_i); the control’s state derivative is multiplied by ε and contributes no additional term at zero. We define the local defects induced by output caching in Appendix C.4 and propagate them using this reference recursion. Cross-time accumulation and statistical visibility. Writing vi=Tieiv_i=T_ie_i, we expand ⟨∑ivi,∑jvj⟩ _iv_i, _jv_j to obtain Eq. (6). The off-diagonal terms retain alignment between transported changes from different steps. Unlike the conditional likelihood scores in Eq. (30) in Appendix A.4 , these output vectors need not be orthogonal in expectation. Adjoint evaluation. For a scalar statistic, we can evaluate the same response by propagating its gradient backward. We define aM=Dϕ(XM)⊤∇f(Y)a_M=Dφ(X_M) ∇ f(Y) and ai=Ji⊤ai+1a_i=J_i a_i+1. Combining this recursion with the tangent recursion, we obtain ai+1⊤δXi+1−ai⊤δXi=ai+1⊤ei,∂εf(Yε)|0=∑iai+1⊤ei.a_i+1 δ X_i+1-a_i δ X_i=a_i+1 e_i, . _ f(Y ) |_0= _ia_i+1 e_i. (25) We derive the second identity by telescoping the first. Each term pairs the local injection with the gradient of the final statistic transported back to that step. For i=μi+LiξiG_i= _i+L_i _i, both tangent and adjoint recursions include the state derivative of LiL_i. With a deterministic time-zero map h0h_0 and decoder D (including latent scaling), we obtain the terminal adjoint aM=Jh0⊤JD⊤∇f(Y)a_M=J_h_0 J_D ∇ f(Y), with each Jacobian evaluated at its corresponding input. These terms ensure that the derivative describes the observed output, including the final transition and decoding. A.4 Path Fisher information and output visibility We next derive the local version of the exact decomposition. Recall that S=∂ϵlogdQϵdP|ϵ=0S= _ε dQ_εdP |_ε=0 is the path likelihood score. We assume the path likelihood admits the expansion Lε=1+εS+rε,‖rε‖L2(P)=o(|ε|),S∈L2(P).L_ =1+ S+r_ , \|r_ \|_L^2(P)=o(| |), S∈ L^2(P). (26) This condition controls the likelihood remainder in mean square and suffices for the response identities below. To obtain second-order KL expansions, we additionally assume that, for some δ>0δ>0 and ε0>0 _0>0, sup0<|ε|<ε0P|Lε−1ε|2+δ<∞. _0<| |< _0E_P | L_ -1 |^2+δ<∞. (27) Normalization gives PS=0E_PS=0. Conditional expectation contracts the L2L^2 remainder in Eq. (26), so Eq. (20) has local score SY=P[S∣Y]S_Y=E_P[S Y]. Because S−SYS-S_Y is orthogonal to every square-integrable function of Y, we obtain PS2=PSY2+P(S−SY)2=FY+PVarP(S∣Y).E_PS^2=E_PS_Y^2+E_P(S-S_Y)^2=F_Y+E_PVar_P(S Y). (28) This proves Eq. (9), where SYS_Y captures the component of the path score resolvable from outputs, while its residual accounts for the unresolved component. For a coarsening Y¯=ψ(Y) Y=ψ(Y), the tower property gives SY¯=P[SY∣Y¯]S_ Y=E_P[S_Y Y]. Using Jensen’s inequality then yields FY¯≤FYF_ Y≤ F_Y. To connect these quantities to KL curvature, we set xε=Lε−1x_ =L_ -1. The function k(x)=(1+x)log(1+x)−xk(x)=(1+x) (1+x)-x, extended continuously at x=−1x=-1, satisfies k(x)=x2/2+o(x2)k(x)=x^2/2+o(x^2) near zero and k(x)≤Cx2k(x)≤ Cx^2 for x≥−1x≥-1. Equations (26)–(27) make the normalized squared increments uniformly integrable. Using Pxε=0E_Px_ =0, we can therefore take expectations of the expansion to obtain DKL(Qε∥P)=12ε2Fpath+o(ε2),DKL(qε,Y∥pY)=12ε2FY+o(ε2),D_KL(Q_ \|P)= 12 ^2F_path+o( ^2), D_KL(q_ ,Y\|p_Y)= 12 ^2F_Y+o( ^2), (29) which is exactly desired form of Eq. (8). Since the output likelihood ratio is the conditional expectation of the path likelihood ratio given the output, conditional Jensen implies that the observed likelihood increments inherit the same moment bound as the path likelihood increments. Subtracting these expansions in Eq. (3), we also obtain bridge|Y(ε)=12ε2Fbridge|Y+o(ε2)C_bridge Y( )= 12 ^2F_bridge Y+o( ^2). Thus FY/FpathF_Y/F_path measures the locally visible fraction whenever Fpath>0F_path>0. Additivity and Gaussian mean perturbations. For differentiable conditional kernels with the same initial law, we define the conditional scores si=∂εlogℓiε(Xi+1,Hi)|0s_i= . _ _i (X_i+1,H_i) |_0 and Fi(Hi)=P[si2∣Hi]F_i(H_i)=E_P[s_i^2 H_i]. Assume that the scores are square integrable and differentiation can be interchanged with conditional normalization. Equation (14) then gives S=∑isiS= _is_i and P[si∣Hi]=0E_P[s_i H_i]=0. For i<ji<j, sis_i is measurable with respect to HjH_j, so P[sisj]=P[siP(sj∣Hj)]=0E_P[s_is_j]=E_P[s_iE_P(s_j H_j)]=0. We therefore obtain Fpath=PS2=∑i=0M−1PFi(Hi).F_path=E_PS^2= _i=0^M-1E_PF_i(H_i). (30) For Gaussian mean perturbations, differentiating Eq. (16) gives si=ui⊤Λi−1(Xi+1−μi)s_i=u_i _i^-1(X_i+1- _i). Using the reference residual’s conditional covariance Λi _i, we obtain Fi(Hi)=ui⊤Λi−1uiF_i(H_i)=u_i _i^-1u_i and hence Fpath=P∑i=0M−1ui⊤Λi−1ui.F_path=E_P _i=0^M-1u_i _i^-1u_i. (31) A.5 Proof of Proposition 3: Feature response and output information We prove Proposition 3 by differentiating the expectation of a statistic. For f(Y)∈L2(P)f(Y)∈ L^2(P), Eq. (26) and Cauchy–Schwarz justify differentiating P[f(Y)Lε]E_P[f(Y)L_ ] at zero. Conditioning on Y and using PSY=0E_PS_Y=0, we derive χf=P[f(Y)S]=P[f(Y)SY]=P[(f(Y)−Pf(Y))SY]. _f=E_P[f(Y)S]=E_P[f(Y)S_Y]=E_P[(f(Y)-E_Pf(Y))S_Y]. (32) Applying Cauchy–Schwarz once more gives χf2≤VarP(f(Y))FY _f^2 _P(f(Y))F_Y. Combining this with Eq. (9) proves the bound in Eq. (10). A.6 Proof of Theorem 1: Perturbation propagation and measured output response We use PSY=0E_PS_Y=0 and Eq. (10) to obtain χf=CovP(f(Y),SY) _f=Cov_P(f(Y),S_Y). For FY>0F_Y>0, we have VarP(SY)=FYVar_P(S_Y)=F_Y and derive BfFY=χf2VarP(f(Y))FY=CorrP(f(Y),SY)2. B_fF_Y= _f^2Var_P(f(Y))F_Y=Corr_P(f(Y),S_Y)^2. (33) Multiplying by FY/FpathF_Y/F_path and substituting the propagated response from Eq. (5) proves Eq. (11). If FY=0F_Y=0, Proposition 3 gives χf=Bf=0 _f=B_f=0, completing the proof. Thus the response depends on the statistic’s alignment with the output score. The centered unit-variance statistic f(Y)=SY/FYf(Y)=S_Y/ F_Y attains equality in the response bound. If FY=0F_Y=0, every square-integrable mean response vanishes. We consequently obtain FY=supPf(Y)=0,Pf(Y)2=1χf2,F_Y= _E_Pf(Y)=0,\;E_Pf(Y)^2=1 _f^2, (34) whenever the class contains a unit-variance observable. A chosen statistic measures one projection of this response geometry; optimizing over all such statistics recovers the full local output information. Fluctuation scale and the limits of a selected mean. Writing Δf(ε)=Qεf(Y)−Pf(Y) _f( )=E_Q_ f(Y)-E_Pf(Y), we have Δf(ε)=εχf+o(ε) _f( )= _f+o( ). Thus ε2Bf ^2B_f is the leading squared mean shift in units of the reference feature variance. For Fpath>0F_path>0, we obtain limε→0Δf(ε)22VarP(f(Y))(ε)=BfFpath. _ → 0 _f( )^22Var_P(f(Y))\,C( )= B_fF_path. Replacing f by af+baf+b with a≠0a≠ 0 multiplies both its squared response and variance by a2a^2, leaving BfB_f unchanged. A smaller absolute response can therefore correspond to a larger BfB_f when the reference variance is sufficiently smaller. The covariance identity also shows that output changes orthogonal to the centered feature have zero first-order mean response, even when FY>0F_Y>0. This concerns the feature’s mean, rather than all changes in its distribution. When FY=0F_Y=0, Eq. (10) directly gives χf=0 _f=0; Eq. (11) is not used because correlation with a zero-variance score is undefined. Predicting finite mean shifts by εχf _f additionally requires the higher-order remainder to be negligible. To connect this statistical derivative with a dynamical derivative, we realize YεY using common initial randomness and innovations, with marginal law qε,Yq_ ,Y. Assume f is continuously differentiable, ∂εYε|0=V _ Y |_0=V exists almost surely, and [f(Yε)−f(Y)]/ε[f(Y )-f(Y)]/ is dominated by an integrable random variable near zero. We exchange differentiation and expectation and apply the chain rule to obtain χf=∂εf(Yε)|0=[∇f(Y)⊤V]. _f= . _ Ef(Y ) |_0=E[∇ f(Y) V]. (35) This is the same distributional mean derivative as the likelihood calculation. Substituting the expansion of V from Eq. (4) yields Eq. (11). Appendix B Gaussian construction for equal-cost perturbations We give the Gaussian reverse-diffusion construction used in Figure 1, in which perturbations with equal path cost reinforce or cancel at the output. Our goal is to hold the noise-normalized intervention strength fixed while changing how its contributions align after propagation. We take C0=diag(1/4,1)C_0=diag(1/4,1) and the forward chain Zs+1=αZs+1−αξsZ_s+1= αZ_s+ 1-α _s, with α=0.9α=0.9, M=24M=24, and independent standard Gaussian innovations. We obtain the forward covariances Cs=αsC0+(1−αs)IC_s=α^sC_0+(1-α^s)I. Conditioning the joint Gaussian law, we derive the exact reverse kernels in generation order, Xi=ZM−iX_i=Z_M-i: Xi+1=AiXi+Λi1/2ξi,Ai=αCs−1Cs−1,Λi=Cs−1−AiCsAi⊤,s=M−i.X_i+1=A_iX_i+ _i^1/2 _i, A_i= αC_s-1C_s^-1, _i=C_s-1-A_iC_sA_i , s=M-i. (36) We start from X0∼(0,CM)X_0 (0,C_M) and observe Y=XMY=X_M, whose reference covariance is C0C_0. We apply deterministic mean controls: Xi+1=AiXi+Λi1/2ξi+εui,ui=(Λi)11wie1,Y=XM.X_i+1=A_iX_i+ _i^1/2 _i+ u_i, u_i= ( _i)_11w_ie_1, Y=X_M. (37) To make the propagation explicit, we define Hi=AM−1⋯Ai+1,hi=(Hi)11(Λi)11,H_i=A_M-1·s A_i+1, h_i=(H_i)_11 ( _i)_11, (38) with HM−1=IH_M-1=I. Here HiH_i transports the injected offset to the endpoint, and hih_i includes its transition-noise scale. Let w=(w0,…,wM−1)⊤w=(w_0,…,w_M-1) be the temporal control vector, and h=(h0,…,hM−1)⊤∈ℝMh=(h_0,…,h_M-1) ^M collect the downstream gains hih_i. We obtain the controlled endpoint mean mY=ε(h⊤w)e1m_Y= (h w)e_1, and its covariance remains C0C_0. Using Eq. (18) and the Gaussian endpoint KL, we derive Fpath=‖w‖2,FY=(h⊤w)2(C0)11,=12ε2‖w‖2,Y=ε2(h⊤w)22(C0)11.F_path=\|w\|^2, F_Y= (h w)^2(C_0)_11, = 12 ^2\|w\|^2, _Y= ^2(h w)^22(C_0)_11. (39) We choose the reinforcing direction w+=h/‖h‖w_+=h/\|h\|. For the cancelling direction, we define ri=(w+)ir_i=(w_+)_i for i<12i<12 and ri=−(w+)ir_i=-(w_+)_i otherwise, and set w−=r−w+(w+⊤r)‖r−w+(w+⊤r)‖.w_-= r-w_+(w_+ r)\|r-w_+(w_+ r)\|. (40) Both directions have unit norm, and h⊤w−=0h w_-=0. Thus the budget depends on the norm of the temporal control, while output change depends on its projection onto the downstream gains. The cancelling control consequently preserves the full endpoint Gaussian law at every amplitude while changing the intermediate path law. At ε=2 =2, the endpoint/bridge KL values are (1.957579,0.042421)(1.957579,0.042421) for the reinforcing control and (0,2)(0,2) for the cancelling control, as shown in Figure 1. The intermediate mean obeys mi+1=Aimi+εuim_i+1=A_im_i+ u_i, with m0=0m_0=0. The observable f(Y)=Y1f(Y)=Y_1 has χf=h⊤w _f=h w and VarP(f(Y))=1/4Var_P(f(Y))=1/4, hence Bf=FYB_f=F_Y in this Gaussian mean-shift family. Appendix C Experimental protocols and measurements C.1 Measuring output responses For each sample, we generate two outputs, Yn+Y_n^+ and Yn−Y_n^-, by applying perturbations of strengths +ε+ and −ε- . We keep the class, initial noise, noise drawn at every sampling step, and spatial template identical in both runs. We then divide the output difference by 2ε2 to estimate the response to the perturbation. For JiT and ADM, we project each generated image onto the DCT template used for the perturbation and average the resulting response over N samples: χ^f=1N∑n=1Nf(Yn+)−f(Yn−)2ε,f(Y)=⟨qk,Y⟩, χ_f= 1N _n=1^N f(Y_n^+)-f(Y_n^-)2 , f(Y)= q_k,Y , (41) where qkq_k is the unit-norm image template defined below. For DiT, we compute the root-mean-square response across the decoded image entries for each sample, then average over samples: R^=1N∑n=1N‖Yn+−Yn−‖2εd,d=3⋅2562. R= 1N _n=1^N \|Y_n^+-Y_n^-\|2 d, d=3· 256^2. (42) For the frequency comparisons in Section 4.4, we estimate the variance sf2s_f^2 of each DCT projection from unperturbed images, using the sample variance with denominator N−1N-1 across classes and noise realizations. We compute B^f=χ^f2/sf2 B_f= χ_f^2/s_f^2, using the same baseline variance for all injection windows of a given feature. We report standard errors of the response means by dividing the sample standard deviation of the per-sample responses by N N. C.2 Reference samplers and perturbation design JiT. The conditional clean predictor Dθ(z,t,c)D_θ(z,t,c) defines drift bt(z,c)=Dθ(z,t,c)−z1−t+λ0(tDθ(z,t,c)−z),σt=2λ0(1−t).b_t(z,c)= D_θ(z,t,c)-z1-t+ _0 (tD_θ(z,t,c)-z ), _t= 2 _0(1-t). (43) The parameter λ0 _0 scales the added diffusion and its matched score drift. We use λ0=0.1 _0=0.1, standard normal pixel initialization, and Euler transitions on ti=iΔt_i=i t, Δt=t∗/128 t=t_*/128, terminating at t∗=0.95t_*=0.95. Their covariance is Λi=νiI _i= _iI with νi=2λ0(1−ti)2Δt _i=2 _0(1-t_i)^2 t. We obtain the generated output as Y=Dθ(Z128,t∗,c)Y=D_θ(Z_128,t_*,c). This stochastic representation is induced by the Gaussian interpolation Zt=tX+(1−t)EZ_t=tX+(1-t)E: with an exact clean predictor, adding score drift λ0(tDθ−z) _0(tD_θ-z) and its matched diffusion preserves the interpolation marginals. With a learned predictor, the displayed kernel specifies the reference law used for the comparisons. ADM. We use the class-conditional ImageNet-256 ADM U-Net. The noise-prediction channels give the clean prediction, clipped to [−1,1][-1,1]. The original 1000-step linear schedule of the denoising diffusion probabilistic model (DDPM) is respaced to 129 retained times: 128 positive-variance transitions with fixed-small posterior covariance, followed by a deterministic time-zero transition. The resulting pixel image is Y. The variance-output channels are unused in this fixed-covariance reference. DiT. DiT-XL/2 evolves a 4×32×324× 32× 32 latent state. The 1000-step linear DDPM schedule is similarly respaced to 129 retained times. The network’s noise and variance outputs specify the transition mean and diagonal learned-range covariance LiLi⊤L_iL_i , with clean-latent clipping disabled. After 128 stochastic transitions, the observed output is Y=DVAE(h0(Z128)/0.18215),Y=D_VAE (h_0(Z_128)/0.18215 ), (44) Here h0h_0 is the deterministic time-zero transition and DVAED_VAE is the variational autoencoder (VAE) decoder. We use the unclamped decoded pixels for response measurements . We compute the output response through the sampler and decoder, including the variance head, h0h_0, and latent scaling. All three model families use conditional predictions without classifier or classifier-free guidance amplification. Spatial templates and temporal normalization. On an H×H× H grid, let ψk(r,s) _k(r,s) be proportional to cos(πk(s+1/2)/H) (π k(s+1/2)/H) and normalized to unit Frobenius norm. These are horizontal modes of the type-I discrete cosine transform: k=0k=0 is the constant DC mode, and k>0k>0 is labeled DCT-k. We use k∈0,1,4,16k∈\0,1,4,16\ for all models and additionally k=31k=31 for DiT. Pixel controls use qk=(1,1,1)⊤⊗ψk/3q_k=(1,1,1) _k/ 3, with H=256H=256. The same template defines fk(Y)=⟨qk,Y⟩f_k(Y)= q_k,Y . For a window B and isotropic transition covariance νiI _iI, both pixel implementations produce the constant mean offset ui=cBqk 1i∈B,cB=(∑i∈Bνi−1)−1/2.u_i=c_Bq_k\,1\i∈ B\, c_B= ( _i∈ B _i^-1 )^-1/2. (45) For JiT this is equivalently a constant drift perturbation cBqk/Δtc_Bq_k/ t within the window. DiT uses H=32H=32, qk=r⊗ψkq_k=r _k with per-sample channel signs rc∈−1/2,1/2r_c∈\-1/2,1/2\, and ui(Zi)=Li(Zi)qk|B| 1i∈B.u_i(Z_i)= L_i(Z_i)q_k |B|\,1\i∈ B\. (46) In both cases the sum of conditional mean-shift Fisher costs is exactly one, giving path KL ε2/2 ^2/2 by Eq. (18). Windows index progress through each model’s sampling schedule. C.3 Models and sampling configurations Table 3: Sample counts N and perturbation amplitudes ε for the controlled perturbation experiments in Section 4. All use 128 stochastic steps, eight injection windows, and path Fisher Fpath=1F_path=1. Model N ε JiT-B/16 512 0.25 JiT-L/16 512 0.25 ADM 256 0.25 DiT-XL/2 256 0.80 We use publicly released class-conditional ImageNet-256 checkpoints for JiT-B/16 and JiT-L/16 [Li and He, 2026], ADM [Dhariwal and Nichol, 2021], and DiT-XL/2 [Peebles and Xie, 2023]. For JiT, we use exponential moving average (EMA) weights. Table 3 lists the sample counts and perturbation amplitudes. We use fixed class subsets, with one noise realization per sample; the reported variability therefore includes both class and noise variation. For comparisons across sampling windows, we keep the classes, initial states, and noise at every step fixed. The DiT local/finite comparison uses 32 paired samples. C.4 Caching protocol and local defects We use the reference kernels and observations in Appendix C.2, with M=64M=64 stochastic steps. JiT uses Δt=0.95/64 t=0.95/64; ADM and DiT use 65 retained diffusion times, comprising 64 stochastic transitions and the final deterministic transition. Each model has 64 paired samples and eight candidate windows B=a,…,a+7B=\a,…,a+7\. For each candidate, we store the network prediction at step a and reuse it directly for the next seven steps, omitting seven network evaluations. Outside B, we evaluate fresh predictions. The final deterministic prediction and decoder are also evaluated afresh. We pair the cached and reference runs using the same class, initial state, and noise at every step. To construct the propagation-based predictor, we evaluate the prediction differences along the reference trajectory. Let oi(Xi)o_i(X_i) denote the fresh prediction at step i. For i∈Bi∈ B, the prediction defect is Δoi=oa(Xa)−oi(Xi) o_i=o_a(X_a)-o_i(X_i); outside B, it is zero. Writing ℋi(x,o,ξ)H_i(x,o,ξ) for the sampling update with a supplied prediction o, we define its local response to this defect by ei=Doℋi(Xi,oi(Xi),ξi)[Δoi].e_i=D_oH_i(X_i,o_i(X_i), _i)[ o_i]. (47) We evaluate this directional derivative at the reference state and prediction, holding the state and noise fixed. The reference Jacobian JiJ_i includes the state dependence of the fresh network prediction. We then propagate the local defects using δZi+1=JiδZi+ei,δZ0=0,V=δY=∑iTiei.δ Z_i+1=J_iδ Z_i+e_i, δ Z_0=0, V=δ Y= _iT_ie_i. (48) At the first step of the window, ea=0e_a=0. We compare ‖V‖2/d\|V\|^2/d with the observed image MSE from direct reuse, as in Eq. (12). This predictor is a first-order approximation along the reference trajectory; its accuracy for the finite caching error is assessed empirically. For JiT, oio_i is the clean prediction and the local update defect is ei=Δt[bi(Zi,oa)−bi(Zi,oi)]e_i= t\,[b_i(Z_i;o_a)-b_i(Z_i;o_i)]. For ADM, oio_i is the clipped clean prediction, the posterior covariance is fixed, and ei=μi(Zi,oa)−μi(Zi,oi)e_i= _i(Z_i;o_a)- _i(Z_i;o_i). Both means are affine in their supplied clean predictions, so these directional derivatives equal the update differences at the fixed reference state. For DiT, we reuse the complete noise-and-variance prediction. Write Σi=LiLi⊤ _i=L_iL_i for its diagonal covariance and ℓi=logdiagΣi _i= _i. At the reference state and prediction, define μ˙i=Doμi[Δoi] μ_i=D_o _i[ o_i] and ℓ˙i=Doℓi[Δoi] _i=D_o _i[ o_i]. The reparameterized step has local response ei=μ˙i+12Lidiag(ℓ˙i)ξi.e_i= μ_i+ 12L_idiag( _i) _i. (49) All state Jacobians include the learned covariance derivative. We propagate the response through the deterministic time-zero map, latent scaling, and VAE decoder. Thus the predicted and observed errors are both measured in decoded pixels. C.5 Caching error prediction and evaluation For each sample and candidate window, we compute RpredR_pred using the tangent recursion. The local-defect baseline is Lstate=∑i‖ei‖2/dzL_state= _i\|e_i\|^2/d_z, where dzd_z is the pixel-state dimension for JiT/ADM and the latent-state dimension for DiT. The path-cost baseline uses the local quadratic approximation Lpath=∑iFi/2L_path= _iF_i/2. At fixed covariance, Fi=ei⊤Σi−1eiF_i=e_i _i^-1e_i. For DiT’s diagonal Gaussian transition, Fi=∑j[μ˙i,j 2Σi,jj+12ℓ˙i,j 2].F_i= _j [ μ_i,j^\,2 _i,j+ 12 _i,j^\,2 ]. (50) Here FiF_i measures conditional Fisher information in the direction of the prediction defect, with the trajectory history held fixed. Its reference expectation summed over steps gives the corresponding path Fisher information. The resulting baseline is a local approximation to path cost, rather than an evaluation of the finite path KL under caching. We average each baseline over samples and compare its ranking across windows with that of observed mean image MSE. C.6 Image visualization protocol The DiT examples use DCT-4 perturbations in windows 4 and 8 with ε=0.8 =0.8, giving path KL 0.320.32 in each window. Images are clipped to [−1,1][-1,1] for display; differences use unclipped decoded pixels. Difference maps show per-pixel RGB RMS relative to the reference, with a common color scale across samples and windows. The scale saturates at the pooled 99.5th percentile. Central-response annotations report the full-image RMS of (Y+−Y−)/(2ε)(Y^+-Y^-)/(2 ). Appendix D Supplementary experimental results D.1 DiT image examples at equal path cost Figures 5 and 6 show two fixed DiT samples under DCT-4 perturbations in windows 4 and 8. Each perturbation has path cost =0.32C=0.32 at ε=0.8 =0.8. The central-response RMS is 19.88×19.88× and 11.23×11.23× larger in window 4 than in window 8 for the two samples, respectively, illustrating the dependence of output response on perturbation timing at equal path cost. Both examples use 128 stochastic steps and the display protocol in Appendix C.6. Figure 5: DiT image responses to equal-cost perturbations. A fixed sample under positive and negative DCT-4 perturbations in windows 4 and 8. Difference maps show per-pixel RGB RMS relative to the reference Y0Y^0, with full-image RMS below each map. Central-response annotations measure the RMS of (Y+−Y−)/(2ε)(Y^+-Y^-)/(2 ); the window-4/window-8 ratio is 19.8819.88. Differences use unclipped pixels and a shared color scale across both examples. Figure 6: Second DiT example, with the endpoints, branch-difference maps, and shared color scale defined in Appendix C.6. Reference-relative displacement RMS is annotated separately for each branch; the window-4/window-8 central-response RMS ratio is 11.2311.23. D.2 Feature responses across sampling windows Tables 4 and 5 extend Table 1 to all eight sampling windows. For each feature, baseline variance is fixed across windows, so the temporal variation in B^f B_f comes from its mean response. Across features, the different baseline fluctuations determine how absolute responses translate into normalized responses. Table 4: Feature mean responses across sampling windows. Signed mean responses χ^f χ_f for windows W1–W8, ordered from early to late sampling, with baseline standard deviations sfs_f. All use ε=0.25 =0.25, with N=512N=512 for JiT-B/L and 256256 for ADM. Model Feature sfs_f W1 W2 W3 W4 W5 W6 W7 W8 JiT-B DC 127127 3.313.31 0.7040.704 0.3610.361 0.2120.212 0.1280.128 0.07860.0786 0.04620.0462 0.01870.0187 DCT-1 48.548.5 2.412.41 0.6610.661 0.3450.345 0.2060.206 0.1320.132 0.07850.0785 0.04420.0442 0.01310.0131 DCT-4 18.918.9 1.431.43 0.5380.538 0.3030.303 0.1890.189 0.120.12 0.07680.0768 0.04070.0407 0.01640.0164 DCT-16 3.883.88 0.3840.384 0.2640.264 0.1830.183 0.1310.131 0.09080.0908 0.0610.061 0.03710.0371 0.01490.0149 JiT-L DC 114114 3.363.36 0.7090.709 0.3580.358 0.2150.215 0.1320.132 0.07830.0783 0.04240.0424 0.01650.0165 DCT-1 45.245.2 2.462.46 0.670.67 0.3510.351 0.210.21 0.1290.129 0.08210.0821 0.04320.0432 0.01680.0168 DCT-4 21.421.4 1.461.46 0.5480.548 0.3060.306 0.1910.191 0.1230.123 0.07730.0773 0.04430.0443 0.01570.0157 DCT-16 4.344.34 0.3660.366 0.2590.259 0.1830.183 0.1310.131 0.09250.0925 0.0620.062 0.03770.0377 0.01510.0151 ADM DC 106106 70.770.7 35.135.1 13.813.8 5.745.74 2.712.71 1.421.42 0.7820.782 0.140.14 DCT-1 42.542.5 24.924.9 22.422.4 1111 5.075.07 2.532.53 1.361.36 0.7640.764 0.1380.138 DCT-4 23.523.5 6.96.9 9.499.49 6.536.53 3.623.62 2.032.03 1.191.19 0.7060.706 0.1350.135 DCT-16 4.324.32 0.2740.274 0.6620.662 1.041.04 1.111.11 0.9120.912 0.6860.686 0.5010.501 0.1180.118 Table 5: Feature responses on the baseline fluctuation scale. Normalized responses B^f=χ^f2/sf2 B_f= χ_f^2/s_f^2 for the measurements in Table 4. Model Feature W1 W2 W3 W4 W5 W6 W7 W8 JiT-B DC 0.0006770.000677 3.06e−53.06e-5 8.05e−68.05e-6 2.77e−62.77e-6 1.01e−61.01e-6 3.82e−73.82e-7 1.32e−71.32e-7 2.17e−82.17e-8 DCT-1 0.002480.00248 0.0001860.000186 5.07e−55.07e-5 1.81e−51.81e-5 7.38e−67.38e-6 2.62e−62.62e-6 8.32e−78.32e-7 7.26e−87.26e-8 DCT-4 0.005770.00577 0.0008130.000813 0.0002580.000258 0.0001010.000101 4.08e−54.08e-5 1.66e−51.66e-5 4.66e−64.66e-6 7.52e−77.52e-7 DCT-16 0.009820.00982 0.004650.00465 0.002220.00222 0.001140.00114 0.0005480.000548 0.0002470.000247 9.13e−59.13e-5 1.48e−51.48e-5 JiT-L DC 0.0008750.000875 3.89e−53.89e-5 9.92e−69.92e-6 3.56e−63.56e-6 1.35e−61.35e-6 4.73e−74.73e-7 1.39e−71.39e-7 2.1e−82.1e-8 DCT-1 0.002960.00296 0.000220.00022 6.03e−56.03e-5 2.16e−52.16e-5 8.2e−68.2e-6 3.3e−63.3e-6 9.14e−79.14e-7 1.39e−71.39e-7 DCT-4 0.004650.00465 0.0006530.000653 0.0002040.000204 7.98e−57.98e-5 3.3e−53.3e-5 1.3e−51.3e-5 4.28e−64.28e-6 5.34e−75.34e-7 DCT-16 0.007120.00712 0.003560.00356 0.001770.00177 0.0009120.000912 0.0004540.000454 0.0002040.000204 7.57e−57.57e-5 1.21e−51.21e-5 ADM DC 0.4470.447 0.110.11 0.0170.017 0.002950.00295 0.0006550.000655 0.000180.00018 5.47e−55.47e-5 1.75e−61.75e-6 DCT-1 0.3430.343 0.2770.277 0.06680.0668 0.01430.0143 0.003540.00354 0.001030.00103 0.0003240.000324 1.06e−51.06e-5 DCT-4 0.08650.0865 0.1640.164 0.07740.0774 0.02390.0239 0.007490.00749 0.002550.00255 0.0009060.000906 3.3e−53.3e-5 DCT-16 0.004020.00402 0.02350.0235 0.05850.0585 0.06610.0661 0.04460.0446 0.02520.0252 0.01340.0134 0.000740.00074