Asymptotic-Preserving A Posteriori Analysis of Diffusion and Flow-Matching Samplers

Jul 16, 2026

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Diffusion and flow-matching samplers integrate a learned probability-flow ODE from a large noise scale to a small terminal floor σmin where the score becomes stiff and forms a boundary layer. Treating σmin as a singular perturbation, the work characterizes fixed-step schemes that are asymptotic-preserving: stable and uniformly accurate as σmin→0. Criteria are expressed as a posteriori audit residuals with σmin-uniform coefficients computable on a pretrained checkpoint, without ground-truth scores or exact trajectories. Terminal-layer rigidity identifies unique layer-exact discretizations for specific clocks, and analyzes where log(1/σmin) cost arises.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages21

Reading Time53 min

Introduction

Preliminaries: One Flow, Many Samplers

Asymptotic-Preserving Terminal-Layer Analysis

Accuracy Across Interfaces and Logarithmic Costs

Predictive Calibration on EDM CIFAR-10

arXiv :2607 .04 1 13v 1 [ cs .LG] 5 Jul 2026 ASYMPTOTIC-PRESERVING A POSTERIORI ANALYSIS OF DIFFUSION AND FLOW-MATCHING SAMPLERS Shiheng Zhang University of Washington [shzhang3@uw.edu](shzhang3@uw.edu) ABSTRACT Diffusion and flow-matching samplers integrate a learned probability-flow ODE from a large noise scale down to a small terminal floor σmin , at which the score is stiff and the flow develops a boundary layer. We treat σmin as a singularperturbation parameter and determine which fixed-step samplers are asymptoticpreserving (AP)—stable and uniformly accurate as σmin → 0—casting the criteria as an a posteriori audit: residual functionals with σmin-uniform coefficients, computable on a pretrained checkpoint without ground-truth scores or exact trajectories. On the terminal layer, Euler in the σ-clock, the deterministic DDIM update, is the unique layer-exact discretization up to affine reparameterization, with rectified flow its flow-matching counterpart; the λ-clock is stable only for steps h ≤ h⋆ = 1 + W(1/e), and the uniform-σ2 heat clock stalls a σmin-independent distance from the data. On two solvable models (rank-deficient Gaussian, symmetric two-point mixture), deterministic samplers remain first-order uniformly accurate with no log(1/σmin) factor, even across a symmetric posterior-switching interface whose distributional budget is a universal constant; the logarithm is charged entirely to the It term of stochastic samplers, whose path-KL scales as Λ2 /N against the ODE’s O(Λ2 /N 2 ) budget, Λ = log(σmax /σmin) . On the EDMCIFAR-10 checkpoint, spectra measured once predict held-out residual budgets across step count, schedule, and noise level against pre-specified gates with no per-configuration refitting, and calibrate the It coefficient at M1 = 1 .00 ± 0.01. The clock decides stability; the noise, not the geometry, charges the logarithm. 1 INTRODUCTION Diffusion and flow-matching models sample by integrating a learned field from noise down to data, stopping at a small but nonzero terminal noise scale: EDM (Karras et al., 2022) uses σmin = 0 .002. The stop is structural: as σmin → 0 the score becomes stiff, its Lipschitz constant growing like σ−m2in (2 .5 × 105 at the EDM floor) . The floor is therefore a genuine singular parameter of the sampling problem, yet it is typically tuned by hand and excluded from the analysis, which either assumes a time-uniformly Lipschitz score or studies an early-stopped process whose iteration count grows like log(1/σmin) (Chen et al., 2022; Benton et al., 2024; De Bortoli, 2022) . This paper analyzes the floor directly. Near the data the probability-flow ODE develops a boundary layer, the structure that asymptotic-preserving (AP) numerics was built to handle (Roos et al., 2008; Jin, 2022), and the appropriate criterion for a fixed-step sampler is stability and accuracy uniform in σmin. Written in the denoiser D and its normalized residual η, the samplers in use (deterministic, stochastic, and flow-matching) are a single flow traversed under different choices of clock, gauge, and noise level (§2), so the AP criterion reduces to a classification of traversal choices. Under this classification, empirical design wisdom on schedules, parameterizations, and stochasticity becomes a set of rigidity theorems, packaged as an a posteriori audit whose functionals are computable on apretrained checkpoint (§3) . Stability on the pure terminal layer is characterized first (§4) . A frozen-field Euler step is layerexact if and only if its clock is affine in σ, which makes the σ-clock step, precisely DDIM (Songet al., 2020a), the unique fitted operator up to affine reparameterization, with rectified flow its flowmatching counterpart; the λ-clock is stable only below the Lambert-W threshold h⋆ = 1+W(1/e); and the uniform-σ2 heat clock cannot resolve the layer at all, stalling a σmin-independent distance from the data. Accuracy beyond the linear regime follows. On the rank-defici

FAQs

What does “asymptotic-preserving” mean for diffusion and flow-matching samplers?

It means a fixed-step sampler remains stable and uniformly accurate as the terminal noise floor σmin approaches 0. The analysis treats σmin as a singular perturbation and checks uniform behavior using audit functionals.

How are asymptotic-preserving criteria evaluated without ground-truth scores or exact trajectories?

The paper reformulates the AP requirement as an a posteriori audit using residual functionals with σmin-uniform coefficients. These residuals are computable from a pretrained checkpoint.

Where does the log(1/σmin) cost arise according to the paper?

The logarithmic cost is charged entirely to the It term of stochastic samplers. Deterministic samplers avoid this logarithmic factor on the studied models, including across a symmetric posterior-switching interface.

Jul 16, 2026

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