{"ID":5937603,"CreatedAt":"2026-07-07T03:14:33.014478982Z","UpdatedAt":"2026-07-08T04:35:57.685350303Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.04113","arxiv_id":"2607.04113","title":"Asymptotic-Preserving A Posteriori Analysis of Diffusion and Flow-Matching Samplers","abstract":"Diffusion and flow-matching samplers integrate a learned probability-flow ODE from a large noise scale down to a small terminal floor $σ_{\\min}$, at which the score is stiff and the flow develops a boundary layer. We treat $σ_{\\min}$ as a singular-perturbation parameter and determine which fixed-step samplers are asymptotic-preserving (AP), that is, stable and uniformly accurate as $σ_{\\min}\\to0$, casting the criteria as an a posteriori audit: residual functionals with $σ_{\\min}$-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 $h\\le h_\\star=1+W(1/e)$, and the uniform-$σ^2$ heat clock stalls a $σ_{\\min}$-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 $\\log(1/σ_{\\min})$ factor, even across a symmetric posterior-switching interface whose distributional budget is a universal constant; the logarithm is charged entirely to the Itô term of stochastic samplers, whose path-KL scales as $Λ^2/N$ against the ODE's $O(Λ^2/N^2)$ budget, with $Λ=\\log(σ_{\\max}/σ_{\\min})$. 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 Itô coefficient at $M_1=1.00\\pm0.01$. The clock decides stability; the noise, not the geometry, charges the logarithm.","short_abstract":"Diffusion and flow-matching samplers integrate a learned probability-flow ODE from a large noise scale down to a small terminal floor $σ_{\\min}$, at which the score is stiff and the flow develops a boundary layer. We treat $σ_{\\min}$ as a singular-perturbation parameter and determine which fixed-step samplers are asymp...","url_abs":"https://arxiv.org/abs/2607.04113","url_pdf":"https://arxiv.org/pdf/2607.04113v1","authors":"[\"Shiheng Zhang\"]","published":"2026-07-05T04:48:33Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"math.NA\"]","methods":"[\"Diffusion Model\"]","has_code":false}