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Universal Local Error and Realized Amplification for the First-Order EDM Predictor

Nicolas Brosse, Arnak S. Dalalyan

arXiv:2610.10190v1stat.MLcs.LG

TL;DR

Deterministic diffusion-sampler accuracy depends both on local discretization and on how later learned steps propagate its error. The paper bounds local error universally and analyzes realized amplification, showing that it accounts for slower error decay on tested finite grids.

  • Problem

    Understanding deterministic diffusion-sampler accuracy requires accounting for both stepwise discretization error and how subsequent steps propagate it.

    Source: §1 · p. 1

  • Method

    The analysis bounds local error through Gaussian smoothing and separates it from learned-step propagation, using EDM preconditioning for high-noise contraction and realized amplification at lower noise.

    Source: §1 · p. 2, §5 · p. 8

  • Results

    On the tested Gaussian mixture, summed local biases approach first-order decay while final error decreases at orders 0.58–0.60; measured amplification recovers the slower decay.

    Source: §1 · p. 1

  • Takeaways & Limitations

    On finite grids, local-bias decay alone may not describe final error; accounting for realized amplification captures the slower decay observed in the mixture experiment.

    Source: §8 · p. 13

    Propagation is estimated a posteriori, and CIFAR-10 diagnostics do not certify the global stability assumptions.

    Source: §8 · p. 13

Abstract

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We analyze the first-order deterministic diffusion sampler of Karras et al. (2022), termed EDM, in 2-Wasserstein distance by separating two sources of error: local discretization error and its amplification by subsequent learned steps. We prove that local error admits a universal bound: for any data distribution with finite second moment, the one-step discretization error is quadratic in the step size, with an explicit constant that does not depend on the data distribution. Error propagation, in contrast, depends on the learned network. At high noise levels, we exploit the network parametrization of EDM to derive an explicit contraction criterion. At low noise levels, we measure propagation through the amplification realized on the distributions transported by the sampler; this realized amplification can be arbitrarily smaller than the worst-case Lipschitz constant. This analysis yields an $O(e^{\Lambda_K}/K)$ global discretization error for $K$ sampling steps, where $\Lambda_K$ is the low-noise log-amplification. Experiments on a one-dimensional Gaussian mixture show how measured amplification accounts for slower error decay on finite sampling grids. Diagnostics on a pretrained CIFAR-10 model illustrate related stability mechanisms without certifying the global assumptions.

1 Introduction

The paper separates local discretization error from its propagation to clarify deterministic diffusion-sampler accuracy. It establishes a distribution-independent local bound and analyzes how learned dynamics amplify errors, including the finite-grid gap between local and final error.

1 Introduction

  • On the tested Gaussian mixture, local biases approach first-order decay while final error decays at orders 0.58–0.60; measured amplification recovers the slower rate.The propagation estimate is a posteriori, CIFAR-10 diagnostics do not certify global stability, and the analysis excludes Heun corrections and stochastic churn.Source: §1 · pp. 1–2, Figure 1 · p. 2
  • Existing Wasserstein analyses use assumptions such as bounded support, strong log-concavity, or geometric conditions; this paper asks whether local error can be bounded without data-law assumptions.It also asks whether worst-case stability estimates overstate propagation.Source: §1 · p. 1
  • For any finite-second-moment data law, the one-step discretization error is quadratic in step size, without support, smoothness, or log-concavity assumptions.The bound follows by controlling conditional Gaussian-noise moments independently of the data distribution's geometry.Source: §1 · p. 2
  • At high noise, EDM preconditioning gives a sufficient contraction condition; at lower noise, realized amplification measures expansion along an optimal coupling of the transported laws.A smooth construction shows realized amplification can remain nonexpansive while worst-case majorants become arbitrarily large.Source: §1 · p. 2
  • The global discretization contribution is O(K^-1 e^ΛK), where ΛK accumulates positive log-amplifications at low noise; high-noise contraction damps early errors.The learning contribution is expressed through excess risk in a normalized regression problem.Source: §1 · p. 2

2 Setting and One-Step Decomposition

The section defines the heat-smoothed laws and exact probability flow that the sampler approximates. It then expresses each first-order EDM step and separates local discretization bias from propagation and denoiser approximation.

2 Setting and One-Step Decomposition

  • The target law is smoothed by adding independent Gaussian noise, giving Xσ = X0 + σZ and a curve of laws transported by the exact probability flow.The flow maps between positive noise levels are invertible and compose across intermediate levels.Source: §2 · p. 3
  • The first-order oracle and learned EDM predictors freeze the velocity at the starting noise level and use the corresponding exact or learned denoiser.Their maps are Φj(x) = (1 − aj)x + ajDσj(x) and its learned-denoiser counterpart.Source: §2 · p. 3
  • The analysis targets the final sampler error and decomposes each step into local discretization bias, denoiser approximation, and propagation of the existing error.Figure 2 distinguishes pushforwards (solid arrows) from W2 distances (dashed links).Source: §2 · pp. 3–4, Figure 2 · p. 4

3 Universal Local Error

This section bounds the local bias through acceleration of the exact flow in EDM’s numerical clock. The resulting bound is quadratic in the clock step and depends on the target law only through dimension.

3 Universal Local Error

  • The refinement family holds noise endpoints fixed and uses nested grids uniform in the numerical clock η = σ^ρ.The commonly used EDM choice ρEDM = 7 corresponds to ρ = 1/7.Source: §3 · pp. 4–5
  • The acceleration bound controls second-order remainders along the exact flow, using conditional Gaussian-noise moments bounded independently of the target law.The noise exponent is sharp over target distributions, while sharpness of the Oρ(d3/2) dimension dependence remains open.Source: §3 · p. 5
  • For any finite-second-moment target and any noise schedule, the one-step local bias has a quadratic clock-step bound whose target-law dependence is only through dimension.The result requires no support, smoothness, or log-concavity assumptions on the data law.Source: §3 · pp. 5–6
  • With ρ = 1/2, the leading local-bias bound is independent of noise and balanced across noise scales, but this need not minimize measured bias on practical grids.For ρ = 1/7, the leading bound instead contains σ^(5/7).Source: §3 · p. 6
  • The local-bias bound is largest at high noise, where the contraction mechanism can reduce early biases’ contribution to final error.Source: §3 · p. 6

4 EDM Preconditioning

EDM preconditioning recasts denoiser learning error as normalized regression error and supplies a sufficient contraction criterion. The resulting recursion separates learning error from contraction and does not itself require contraction.

4 EDM Preconditioning

  • EDM preconditioning expresses the learning contribution through the residual of a normalized regression problem, with the residual controlling the learning term.The normalized target has unit second-moment scale relative to dimension.Source: §4 · pp. 6–7
  • The prediction loss upper-bounds the normalized excess risk, but the unknown Bayes risk prevents recovering that excess risk from prediction loss alone.Checking that the residual remains small across all noise levels is difficult in practice.Source: §4 · p. 7
  • A positive EDM stability margin certifies strict contraction when the normalized network Lipschitz constant is below σj/τq; higher noise permits larger constants.A nonpositive margin does not certify contraction, though the learned predictor may still contract the transported laws.Source: §4 · p. 7
  • The assembled recursion bounds the learning term using the normalized residual, while the recursion itself requires no contraction assumption.The later propagation analysis combines high-noise contraction with amplification over remaining steps.Source: §4 · p. 7

5 High-Noise Contraction and Realized Amplification

The learned sampler contracts at high noise under an explicit sufficient condition, while low-noise error propagation is tracked through realized amplification. Directional and worst-case bounds control that amplification, but worst-case majorants can greatly exceed what the transported laws realize.

High-noise contraction

  • A high-noise network with a uniform normalized-input Lipschitz bound and EDM margin above a fixed positive threshold satisfies the transport contraction condition.For the oracle with data supported in a radius-R ball, the transport condition holds above σhi = R/√(1 − bhi).Source: §5 · p. 8

Realized amplification

  • Low-noise propagation accumulates the logarithms of realized amplification factors, whose exponent bounds propagation across any portion of the low-noise block.The realized factors are defined on transported laws; no contraction assumption is imposed on the remaining steps.Source: §5 · p. 8

Propagation bounds

  • The directional bound measures expansion along an optimal input coupling, while global map and field bounds maximize over space rather than track the particular laws.The directional transported coupling need not remain optimal after the learned map, so its cost upper-bounds the Wasserstein distance.Source: §5 · pp. 8–9

Separation from worst-case stability

  • A smooth construction keeps realized amplification nonexpansive while worst-case majorants become arbitrarily large as ε tends to zero.The separation uses an O(ε) oscillation with slope O(ε^-1) and an O(ε) initialization translation; for each fixed ε, the majorants have finite refinement limits.Source: §5 · p. 9
  • The separation construction uses an alternative initialization fixed across refinements for each ε, rather than the section’s Gaussian initialization.The initialization choice is essential to this example.Source: §5 · p. 9

6 Global Error Bound

The global bound separates initialization, discretization, learning, and terminal smoothing contributions. At fixed positive endpoints, the discretization term scales with the step size and accumulated low-noise realized amplification; high-noise contraction damps early biases.

Global bound

  • The global bound separates initialization, discretization, learning, and terminal smoothing bias into four contributions.The discretization term includes both undamped low-noise and damped high-noise local biases.Source: §6 · p. 10
  • At fixed endpoints, the discretization contribution is O(hK e^ΛK), with high-noise contraction reducing its prefactor by damping early biases.For fixed endpoints and clock, hK = O(K^-1).Source: §6 · p. 10

Rates under refinement

  • When ΛK grows at most logarithmically as O(ω log K), the discretization term is O(K^-(1−ω)); a uniform bound on ΛK gives first-order decay.For ρ = 1/7, the stated high-noise scale condition is beff > 6/7.Source: §6 · p. 10

Scope

  • Refinement does not remove learning error, initialization mismatch, or terminal smoothing bias, and first-order behavior may emerge only beyond practical step counts.The logarithmic-growth envelope is an asymptotic assumption, not a consequence of a finite-range fit.Source: §6 · p. 10

7 Numerical Experiments

On a Gaussian mixture, measured amplification explains why final error decays more slowly than accumulated local bias on finite grids. Further experiments compare directional bounds with worst-case proxies and probe related stability mechanisms in a pretrained CIFAR–10 model without certifying global assumptions.

7.1 Amplification and the Finite-Grid Rate

  • The summed local biases approach first-order decay, but final Wasserstein error has orders 0.58–0.60; measured amplification accounts for this slower decay on the tested grids.The amplification-based order agrees with observed error orders on the two finest grid pairs.Source: §7.1 · p. 11
  • For the oracle, ΛK is uniformly bounded under refinement, so the slower observed decay is preasymptotic and the eventual bound is first-order.The measured amplification is computed a posteriori from transported laws rather than predicted from the denoiser.Source: §7.1 · p. 11

7.2 Directional Bounds Track Realized Amplification

  • At K = 1088, the directional majorant exceeds ΛK by at most 2.2% for every denoiser, while the oracle field proxy is 143 times ΛK on the coarsest grid and 11.1 times on the finest.The differences become more pronounced after exponentiation in the global error bound.Source: §7.2 · p. 12
  • The oracle’s velocity expands in narrow bands between modes, while the discrepancy lies mostly in contracting regions, making the field supremum looser than the coupling-weighted directional bound.The field supremum includes expanding regions regardless of where the discrepancy lies.Source: §7.2 · p. 12

7.3 Stability Mechanisms in an Image Model

  • The empirical high-noise margin probe becomes positive at σ = 8.9 and exceeds 6/7 above σ = 40.8, but it does not establish a global Lipschitz bound or valid contraction threshold.The diagnostics use a pretrained class-conditional CIFAR–10 EDM checkpoint.Source: §7.3 · p. 12
  • At σ = 0.004, random directions contract at λrand = −82 while displacement expands at λ∥ = +0.49; the worst-direction rate is 424 times the displacement rate.These CIFAR–10 rates are local diagnostics along synchronously coupled coarse and fine trajectories, not estimates of ΛK.Source: §7.3 · p. 13

8 Discussion and Limitations

The global estimate separates universal local discretization error from its amplification by learned dynamics, a distinction that matters on finite sampling grids. Realized amplification remains a posteriori, and the CIFAR-10 diagnostics do not certify global stability assumptions.

8 Discussion and Limitations

  • On finite grids, accumulated local biases and final Wasserstein error can decrease at different rates, so local accuracy alone may not describe sampler behavior.The discussion identifies measured amplification as the link between these distinct decay rates.Source: §8 · p. 13
  • Propagation remains a posteriori, CIFAR-10 diagnostics do not certify global stability, and the O(d^3/2) local constant limits quantitative use in high dimensions.A computable propagation certificate retaining distribution-dependent control is identified as a central next step.Source: §8 · p. 13

AI Use Statement

The authors disclose their use of generative AI and responsibility for the paper’s content, while the appendices establish notation, posterior and exact-flow foundations, and refinement-family conditions. They also distinguish the finite-step EDM predictor from explicit η-Euler in terms of clock dependence and coarse-grid behavior.

AI Use Statement

  • Generative AI assisted with proofs, experiments, and writing, while the authors retained the research direction, reviewed results, and took responsibility for the paper.Figures, tables, and reported measurements were generated from saved experiment outputs in the public code repository.Source: AI Use Statement · p. 13

Appendix Contents

  • Appendix A collects notation; Appendices B–G provide proofs aligned with Sections 2–7, and Appendix H compares the analysis with related Wasserstein work.Source: Appendix Contents · p. 18

A Notation Index

  • The notation index defines general and problem-specific symbols, including coupling superscripts, grid indices, and markers distinguishing learned objects from exact counterparts.Source: A Notation Index · p. 18

B Setting and One-Step Decomposition

  • Appendix B constructs the exact probability flow, specifies the clock and refinement family, relates the EDM predictor to explicit Euler, and proves the one-step decomposition.It also compares that decomposition with two alternative orderings.Source: B Setting and One-Step Decomposition · p. 20

B.1 Posterior Identities and the Exact σ-Flow

  • Appendix B.1 establishes posterior identities and exact-flow properties for every Borel data law, using positive-noise intervals without assuming moments of the data law.Source: §B.1 · p. 20

B.1.1 Gaussian Smoothing and Posterior Moment Identities

  • Gaussian smoothing yields a positive smooth density and a posterior kernel valid pointwise, with finite posterior moments and conditional identities that also cover unbounded integrands.Source: §B.1.1 · pp. 20–21

B.1.2 Tweedie’s Formula and Linear Growth of the Denoiser

  • Tweedie’s identity equates the score-based velocity with the denoiser residual, and the denoiser and velocity have linear growth on compact positive-noise intervals.Source: §B.1.2 · p. 23

B.1.3 The Continuity Equation and the Exact Flow Maps

  • The posterior velocity satisfies the continuity equation and generates unique exact flow maps on compact positive-noise intervals.Source: §B.1.3 · pp. 24–25
  • The exact flow maps are smooth diffeomorphisms that transport the heat-regularized laws, and pushing forward any finite-second-moment law preserves finite second moment.The transport identity holds for all noise-level pairs.Source: §B.1.3 · pp. 25–26

B.2 The Refinement Family

  • The refinement family must be infinite and use fixed endpoints; uniform η steps have their largest relative size at the smallest η, where the schedule-wide bound applies.Fixed endpoints keep the terminal bias unchanged across grids and allow schedule-uniform per-step bounds to be summed into a schedule-uniform total.Source: §B.2 · p. 26

B.3 From η-Euler to the EDM Predictor

  • EDM and explicit η-Euler use the same denoiser and noise levels but different finite-step coefficients; they agree to first order and may diverge on coarse terminal grids.The EDM predictor is invariant across power-law clocks, whereas explicit η-Euler is not; the comparison concerns the deterministic first-order predictor, not stochastic churn or a Heun correction.Source: §B.3 · pp. 26–27

B.4 Proof of the One-Step Decomposition

The one-step decomposition keeps discretization and learning errors on the exact law while isolating propagation through the learned map. The appendix derives a distribution-independent quadratic local-bias bound and explains how clock choice affects its scale dependence.

B.4 Proof of the One-Step Decomposition

  • The decomposition bounds discretization and learning errors on the exact law, with the learned map’s stability controlling propagation.The third term is the propagated error; the first two are measured on the exact law.Source: §B.4 · p. 28

B.5 Three One-Step Decompositions

  • Among three decompositions, the chosen coupling-level form keeps both local errors on the exact law and puts network dependence in learned-map stability.Alternative choices move a local error off the exact law or incur a stability cost from the oracle predictor.Source: §B.5 · pp. 28–29

C.1 Clock Change and the Score–Hessian Form

  • A curved clock adds a velocity term to the acceleration, whereas the linear clock does not.For η = σ^ρ, the added term has coefficient proportional to 1 − ρ.Source: §C.1 · p. 29

C.2 The Noise-Moment Identity

  • The noise-moment identity rewrites clock acceleration using conditional Gaussian-noise moments, providing the pivot for the universal bound and exact Gaussian formulas.The score, Hessian, and gradient of its trace are replaced by conditional noise moments.Source: §C.2 · pp. 31, 33

C.3 Proof of the Universal Bound

  • Bounding conditional means, covariances, and third moments by Gaussian moments yields a distribution-independent acceleration bound with sharp noise exponent.The universal bound’s dimension dependence is Oρ(d^3/2), whose sharpness remains open.Source: §C.3 · p. 34

C.4 Proof of the Local Bias Bound

  • The local bias is the Taylor remainder between an exact flow step and the oracle predictor, controlled by the native-clock acceleration bound.The oracle predictor freezes the σ-clock velocity at the step’s left endpoint.Source: §C.4 · p. 35

C.5 The Leading-Order Local Bias and the Clock

  • The clock ρ = 1/2 (ρEDM = 2) equalizes the leading local-bias bound across noise scales, but is not claimed to optimize schedules or final sample quality.The clock experiment finds that balancing the universal bound does not yield the smallest measured biases or final error.Source: §C.5 · pp. 35–36

C.6 Gaussian Laws and Sharpness

Gaussian inputs permit exact evaluation of the acceleration, while worst-case input laws make the universal noise exponent sharp. The appendix also identifies a dimension-dependent looseness in the bound and explains the role of EDM preconditioning in contraction.

C.6 Gaussian Laws and Sharpness

  • For Gaussian input laws, the acceleration has an exact covariance-spectrum formula, and the universal noise-scale exponent is sharp over all Borel laws.The Gaussian specialization has vanishing centered third conditional moment; singular laws witness the sharp universal exponent.Source: §C.6 · pp. 36–37
  • The universal acceleration bound is dimensionally lossy: whether its dimension dependence can be improved without additional moment assumptions remains open.The current constant at CIFAR–10 dimension d = 3,072 is approximately 6.8 × 10^5.Source: §C.6 · pp. 37–39

D.1 Complements on the EDM Margin

  • The EDM margin implies a one-sided monotonicity property of the learned velocity, while its affine predictor representation avoids a finite-step loss from monotonicity alone.A uniform normalized Lipschitz envelope can provide an admissible high-noise threshold.Source: §D.1 · p. 39

D.2 High-Noise Contraction for the Oracle

High-noise contraction for the oracle requires a suitable bound on the data law’s support. For low-noise propagation, directional rates measure expansion along transported discrepancies and can be smaller than global majorants; integrability can bound their accumulated contribution.

D.2 High-Noise Contraction for the Oracle

  • For the oracle, high-noise contraction is not automatic: unbounded support can make the denoiser Jacobian’s supremum infinite, while bounded support yields a threshold certificate.If supp q ⊆ B(0, R), the stated certificate gives σhi = R/√(1 − bhi).Source: §D.2 · p. 39

E.1 The Split Index

  • The split index is the last noise level at or above σhi; the threshold-crossing step starts the low-noise block.Every high-noise step ends at or above σhi.Source: §E.1 · p. 40, Figure 5 · p. 40

E.2 Low-Noise Majorants: Proofs

  • Directional and map-level low-noise majorants are not ordered in general, because each relaxes different information about the realized discrepancy or tested directions.For bv(x) = −x, one step contributes ℓ + ℓ²/2 to the directional sum but only log(1 + ℓ) to the map sum.Source: §E.2 · p. 42

E.3 Geometric Reading of the Directional Rate

  • The directional rate averages expansion along the segments traveled by the optimal transport, so it can be much smaller when displacement avoids expansive directions.The sampled-segment quotients are diagnostics, not certificates of the global one-sided rate.Source: §E.3 · p. 43
  • An integrable global one-sided rate, together with a vanishing finite-step remainder, is sufficient to keep the field-level majorant of accumulated low-noise amplification bounded.For the oracle field, the rates depend on the small-noise behavior of the smoothed data law’s log-Hessian.Source: §E.3 · p. 43

E.4 The Strict Separation from Worst-Case Stability

This section separates realized amplification from worst-case regularity and develops a global Wasserstein bound using high-noise contraction and mesh-weighted bias estimates. It also gives a uniform damped bias-sum bound and a step-size condition for contraction of the high-noise kernel on a finite mesh.

E.4 The Strict Separation from Worst-Case Stability

  • The separation uses translated initialization to keep discrepancy large relative to perturbation amplitude, allowing Gaussian contraction to absorb perturbations; it does not establish the same result for exact initialization and purely discretization-induced error.The construction's translation is costly: for the stated EDM endpoints, the endpoint ratio is 4 × 10^4 and 4/ϖmin is approximately 500.Source: §E.4 · pp. 44, 47
  • Zero cumulative log-amplification can coexist with infinite worst-case constants, while a related construction has finite constants at each fixed perturbation scale that grow arbitrarily large as the scale shrinks.The fixed construction keeps ΛK = 0 while every predictor has infinite global Lipschitz constant and every one-sided rate is infinite.Source: §E.4 · p. 47

F The Global Wasserstein Bound

  • Appendix F proves the high- and low-noise block estimates, combines them for the global bound, and also bounds initialization error and extends the bound to deployed samples.Source: F The Global Wasserstein Bound · p. 47

F.1 Explicit Form of the Global Bound

  • On finite meshes, contraction is reduced by κ(θ), which approaches 1 as θ shrinks; the subcritical and supercritical regimes differ in which noise scale controls the high-noise bias sum.For ρ = 1/7, the supercritical condition is beff > 6/7 and requires bhi > 6/7, met on sufficiently fine meshes.Source: §F.1 · p. 48

F.2 High-Noise Block

  • Under high-noise contraction, the error recursion damps initialization error and the kernel saturates accumulated learning error; the estimate covers 0 < ρ ≤ 1, not ρ > 1.The refinement-family estimate assumes a mesh condition, with the discrete contraction kernel bounded by a power using effective exponent beff.Source: §F.2 · pp. 48–49

F.3 Summation of the Local Biases

  • The summation argument compares mesh sums with integrals, bounds universal bias terms using a power kernel and the scale function Ξρ, and combines these bounds with discrete contraction.Source: §F.3 · p. 49

F.3.1 A Relative-Mesh Riemann Comparison

  • With relative mesh size h ≤ θηj, the Riemann-sum comparison bounds the grid sum of η^-p by a corresponding integral with a θ-dependent cost.Source: §F.3.1 · p. 50

F.3.2 The Damped Bias Sum

  • Under the same relative-mesh condition, the damped universal-bias sum is bounded through Ξρ, with the kernel exponent b ranging from 0 to 1.Source: §F.3.2 · p. 50
  • As θ decreases to zero, the bias-sum constant tends to 1/(2ρ); it is uniform over b in [0, 1] and also covers the undamped sum at b = 0.The same lemma applies to both the damped high-noise sum and the undamped low-noise sum.Source: §F.3.2 · p. 50

F.3.3 Proof of the High-Noise Estimate

  • If h ≤ θη_i on the segment, the finite-mesh kernel satisfies Π_j,k ≤ (σ_k/σ_j)^bhi κ(θ).The proof bounds discrete log-noise relaxation using a_i = 1 − e^-r_i ≤ r_i, with r_i controlled by h/η_i ≤ θ.Source: §F.3.3 · p. 51

F.4 Low-Noise Block and Proof of the Global Bound

The low-noise block closes the error recursion using cumulative realized amplification, while the global proof combines low- and high-noise discretization terms. The corollary relates the resulting rate to growth in the amplification envelope.

F.4 Low-Noise Block and Proof of the Global Bound

  • Without a contraction assumption below the threshold, the low-noise recursion bounds partial products by e^ΛK and sums the universal bias and learning contributions.The low-noise estimate applies under Assumption 5 and a mesh condition hK ≤ θloσ^ρ.Source: §F.4 · pp. 51–52
  • The learning scale Υ(s1, s0) is at most min{s0 − s1, τq log(1 + 2s0/τq)}, so it is bounded by the noise-range length and grows only logarithmically for large s0.The passage attributes these bounds to an integrand at most one and the large-argument behavior of arsinh.Source: §F.4 · p. 52
  • The global proof combines high- and low-noise discretization terms into DiscK, with either block's terms vanishing when that block is empty.Theorem 11 then follows by bounding Υ.Source: §F.4 · p. 52
  • Refinement does not improve the denoiser or remove initialization mismatch, and a growing e^ΛK envelope can increase the bound even when actual error stays bounded.For ρ < 1, the low-noise scale-function term has a bound proportional to σKhi^(1−ρ)/(1−ρ).Source: §F.4 · p. 52
  • When e^ΛK = O(K^ω), the oracle with exact initialization has eK = O(K^-(1-ω)).The estimate does not require Assumption 6 when the high-noise block is empty; adding terminal smoothing bias gives the second assertion.Source: §F.4 · p. 52

F.5 Initialization Error at the Top Noise Level

This section bounds top-noise initialization error, including for deployed samples, and examines its relation to refinement and denoiser-dependent amplification. It also reports numerical audits of initialization, amplification, and stability, while noting limits on low-noise segment diagnostics and trajectory-level sensitivity.

F.5 Initialization Error at the Top Noise Level

  • For every σ0 > 0, e0 ≤ sqrt(d)τq; under light tails, 2σ0e0 tends to ||Σdata||F as σ0 grows.The limiting constant is bounded by tr Σdata = dτq^2.Source: §F.5 · pp. 52–53

F.6 From the Positive-Noise Iterate to the Deployed Sample

  • With a globally Lipschitz terminal denoiser, the emitted law is bounded by propagating the positive-noise iterate error through that denoiser.The resulting bound separates propagated sampler, learning, and posterior-denoising errors; the Lipschitz condition is global.Source: §F.6 · pp. 53–54

G.1.1 Shared Schedule and Implementation

  • The shared implementation matches coarse and fine trajectories at shared noise levels and uses the public class-conditional CIFAR–10 EMA checkpoint.The one-dimensional audit uses float64, while CIFAR–10 inference uses float32 with recorded-state subtraction and reductions in float64.Source: §G.1.1 · p. 54

G.1.2 One-Dimensional Audit

  • The one-dimensional audit uses nested EDM grids, float64 calculations, and a shared noise threshold of σhi = 1.84 across denoisers and refinements.The threshold is the largest of candidates 0.876, 1.53, 1.57, and 1.84, and is an empirical split rather than a global certificate. Field proxies are not certified bounds, and the reported error-term ratios are contributions to an a posteriori upper bound, not an error decomposition.Source: §G.1.1 · p. 54, §G.1.2 · pp. 56–57

G.1.3 High-Noise CIFAR–10 Diagnostics

  • CIFAR–10 diagnostics use held-out noisy probes, but the population risk is only an upper proxy for learning residual and the empirical margin is optimistic, not a certificate.The empirical margin replaces a global Lipschitz supremum with probe percentiles and estimated stretches.Source: §G.1.3 · pp. 57–58

G.1.4 Low-Noise Synchronous-Segment Diagnostic

  • The synchronous-segment diagnostic compares realized amplification with segment-based upper bounds using paired coarse and fine trajectories.For continuous quantities, the rate chain is exact; its synchronous rate and segment quantities use the same states and energy weights.Source: §G.1.4 · pp. 58–59
  • The synchronous-segment inequalities are exact for continuous quantities, but finite-node maxima and unconverged Ritz values can underestimate suprema, making computed inequalities empirical consistency checks rather than certificates.The segment calculation requires at most 368,640 Lanczos operator applications, plus two Jacobian–vector products per node for directional and random rates.Source: §G.1.4 · p. 59
  • Bootstrap sensitivity intervals describe sampling rather than Jacobian-approximation error, and the fixed subset may miss influential trajectories; non-finite states or positive-energy rates invalidate aggregates rather than being dropped or set to zero.A reproducible large discrepancy is retained as a measured phenomenon, while unresolved discrepancies are reported as missing or qualified evidence.Source: §G.1.4 · pp. 59–60
  • The accumulated quadratic term stays below 0.09 on every grid, from 0.087 at K = 17 to 0.038 at K = 272; leave-one-out ranges measure single-trajectory influence, not statistical confidence.No cumulative field curve is inferred from the eight sparse segment levels.Source: §G.1.4 · p. 60

G.2 Full Figures and Tables

  • Summed local biases converge at first order, but final error does not; growth of measured ΛK accounts for the gap.Figure 7 reports this pattern for the oracle denoiser on the four-component mixture with exact initialization.Source: Figure 7 · p. 61
  • The directional bound’s excess over ΛK comes from log(1 + x) ≤ x and the positive part; the field proxy’s coarse-grid quadratic term vanishes under refinement, leaving a first-order geometric term.In one dimension, the field rate is the supremum of the local expansion rate, while the directional rate averages it under EOT.Source: §G.2 · p. 61
  • At K = 1088, trained networks stay near the oracle except seed 0, whose ΛK is 1.33 versus 1.88 for the other two seeds; the misspecified posterior mean settles near 0.67.The misspecified model’s error is dominated by learning.Source: §G.2 · p. 61

G.3 Supplementary Results: One-Dimensional Audit

The supplementary audit examines finite-grid amplification and error behavior, then adds analytic controls, CIFAR–10 diagnostics, and numerical reliability checks. Its comparisons also clarify how this analysis differs from prior Wasserstein approaches.

G.3 Supplementary Results: One-Dimensional Audit

  • The oracle’s logarithmic growth and sublinear order are preasymptotic: first-order decay must eventually return, but the uniform bound is quantitatively uninformative.Perturbed oracles with ε ⩽0.03 show isolated ΛK collapses when local bias and perturbation nearly cancel; these are not trends.Source: §G.3 · pp. 63–64
  • With the oracle, the canonical Gaussian start remains discretization-dominated; with trained denoisers, learning overtakes discretization from K = 272 and error levels off near 0.018 for two seeds.The propagated initialization term stays below 6.2×10−5, versus 0.0136 for discretization at K = 1088.Source: §G.3 · p. 64, Figure 10 · p. 64
  • Oracle ΛK grows by 0.22–0.29 per doubling, while fixed-noise rates vary by noise level and large-floor arms converge at every tested level.At σ ≈0.065 the oracle rate rises from 3.3 to 34; the misspecified mean and ε = 0.1 perturbed oracle converge in both ΛK and fixed-noise rates.Source: §G.3 · p. 64

G.4 Analytic Controls

  • The universal bias bound can reverse the observed clock ranking, while high-noise damping reduces accumulated bounds and synchronous rates need not track ΛK for an inexact denoiser.At K = 544, measured biases and final error rank clocks ρEDM = 7 < 3 < 2, opposite the universal bound; increasing starting noise from 20 to 318 multiplies the undamped high-noise sum by 14 but the damped sum by 2.6. For the misspecified denoiser, the synchronous rate rises from 1.11 to 1.87 while the optimal-coupling rate falls from 0.76 to 0.69.Source: §G.4 · pp. 65–66, Table 4 · p. 65, Table 5 · p. 65

G.5 Supplementary CIFAR–10 Diagnostics

  • On CIFAR–10, the energy-weighted rate rises from 0.41 to 0.67 across K = 34, 68, 136, but rare trajectory pairs make the refinement trend ensemble-sensitive.At K = 272, 21 trajectories carry half the leave-one-out change; five pairs have squared displacements over 100 times their class median.Source: §G.5 · p. 66

G.6 Numerical Reliability and Sensitivity Checks

  • Two checks fail on the fine CIFAR segment grid G272, so the corresponding statistics are omitted; quadrature refinement preserves agreement between error and amplification-based orders.Their individual orders are not resolved to the third decimal, but their agreement is robust to the tested refinement.Source: §G.6 · p. 66, Table 8 · p. 68

G.7 Code and Data Availability

  • The public repository provides code and frozen raw outputs, with scripts regenerating the main numerical figures, tables, and quoted measurements.The one-dimensional experiments take a few CPU hours in total; the longest CIFAR–10 diagnostic takes about six hours on one NVIDIA L4 GPU.Source: §G.7 · p. 69

H Comparison with Prior Wasserstein Analyses

  • Compared with prior Wasserstein analyses, this work removes structural assumptions from local bias control but retains learned-dynamics conditions for propagation.Its global theorem covers only the first-order deterministic predictor, unlike Beyler–Bach’s coverage of Heun and Euler–Maruyama.Source: H Comparison with Prior Wasserstein… · p. 69

H.1 Closest Comparison: Beyler–Bach

The comparison contrasts support- or geometry-dependent Wasserstein analyses with a universal local-bias bound and propagation measured on transported laws. It also explains why score accuracy alone does not guarantee sampler stability.

H.1 Closest Comparison: Beyler–Bach

  • Beyler and Bach’s local control depends on bounded support, whereas this paper’s posterior-identity estimate applies to every Borel data law; its propagation factor is sharper but a posteriori.The global theorem here covers only the first-order deterministic predictor, while Beyler–Bach also cover Heun and Euler–Maruyama.Source: §H.1 · pp. 69–70

H.2 A Structural Wasserstein Route: Gao–Zhu

  • Gao–Zhu derive contraction from strong log-concavity, while this analysis requires no such geometry for local bias and instead tracks low-noise cumulative realized amplification.The two analyses separate initialization, discretization, and learning errors in the same metric, but place the structural burden at different points.Source: §H.2 · p. 71

H.3 Score Accuracy Does Not Imply Stability

  • Zhou’s counterexamples show that small score error alone need not ensure stability: explicit Euler can develop diverging moments, while deterministic probability-flow laws can diverge in every Wp.The deterministic example does not contradict this theorem because its field has superlinear tail growth and violates Assumption 4.Source: §H.3 · p. 71

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