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

arXiv:2607.04113v1 Announce Type: new Abstract: Diffusion and flow-matching samplers integrate a learned probability-flow ODE from a large noise scale down to a small terminal floor sigma_{min}, at which the score is stiff and the flow develops a boundary layer. We treat sigma_{min} as a singular-perturbation parameter and determine which fixed-step samplers are asymptotic-preserving (AP), that is, stable and uniformly accurate as sigma_{min}o0, casting the criteria as an a posteriori audit: residual functionals with sigma_{min}-uniform coefficients, computable on a pretrained checkpoint without ground-truth scores or exact trajectories. On the terminal layer, Euler in the sigma-clock, the deterministic DDIM update, is the unique layer-exact discretization up to affine reparameterization, with rectified flow its flow-matching counterpart; the lambda-clock is stable only for steps hle h_star=1+W(1/e), and the uniform-sigma^2 heat clock stalls a sigma_{min}-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/sigma_{min}) factor, even across a symmetric posterior-switching interface whose distributional budget is a universal constant; the logarithm is charged entirely to the Ito term of stochastic samplers, whose path-KL scales as Lambda^2/N against the ODE's O(Lambda^2/N^2) budget, with Lambda=log(sigma_{max}/sigma_{min}). On the EDM CIFAR-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 Ito coefficient at M_1=1.00pm0.01. The clock decides stability; the noise, not the geometry, charges the logarithm.

Source: arXiv cs.LG | 2026-07-07