---
title: "Score Accuracy Along the Forward Diffusion Does Not Certify Numerical Stability in Diffusion Sampling"
source: "arXiv AI/CL/LG（论文）"
source_tier: "t2"
url: "https://arxiv.org/abs/2607.08757v1"
permalink: "https://feed.astralor.com/items/itm_b9050a5597b829187d93e38a"
published: "2026-07-09T17:55:52.000Z"
score: 79
channel: "ai"
tags: ["Diffusion Transformer","Score Matching","Diffusion Models","Numerical Stability","Euler-Maruyama"]
---

# Score Accuracy Along the Forward Diffusion Does Not Certify Numerical Stability in Diffusion Sampling

## AI commentary

> This paper hits the nail on the head: diffusion model sampling may seem stable, but theoretical loopholes can cause sudden 'explosions' in generation. It's a must-read warning for engineers deploying diffusion models in production. The paper not only identifies the problem but also offers a practical remedy (denoiser projection), making it a must-watch for all generative model developers.

## At a glance

This paper demonstrates that small score-matching error under forward marginals does not guarantee numerical stability in discretized reverse sampling. The authors construct smooth score fields with arbitrarily small forward-marginal L² error and path-space total variation, yet Euler–Maruyama discretizations diverge in every Wasserstein distance. For compactly supported data, projecting the denoiser onto a bounded convex set restores stability. Experiments with a small DiT network show large growth on rare trajectories suppressed by projection.

## Key points

- Small forward-marginal score error does not guarantee numerical stability; counterexamples make every Wasserstein distance diverge
- The constructed score field is smooth and the continuous reverse process is correct, but Euler–Maruyama discretization causes moment blow-up
- For compactly supported data, projecting the denoiser onto the convex hull of the support preserves accuracy and yields uniform moment bounds
- Experiments with a small DiT network show numerical explosion on rare trajectories, suppressed by denoiser projection

## Full text

Score matching controls average error under the forward marginals, but a discretized reverse-time sampler evaluates the learned score along its own trajectory. We show that small forward-marginal error does not guarantee numerical stability. We construct a single smooth score field with arbitrarily small forward-marginal $L^2$ error. The learned reverse-time process is nonexplosive, has moments of every order, and can be arbitrarily close to the exact reverse-time process in path-space total variation. Yet its Euler--Maruyama discretizations converge in probability while every positive moment diverges. Thus weak convergence can hold even though every Wasserstein distance $W_p$, $p\ge1$, diverges. The same failure can occur within one fixed finite neural architecture. We construct a family of bounded, globally Lipschitz denoisers for which both the forward-marginal error and the path-space total variation distance tend to zero, while their Euler--Maruyama endpoints diverge in every $W_p$. For compactly supported data, we also give a simple positive result. Projecting the learned denoiser onto a known bounded closed convex set containing the support preserves pointwise accuracy, gives grid-uniform moment bounds, and yields Wasserstein convergence under mild local regularity. Experiments with a small fixed DiT-style network show large growth along rare numerical trajectories and its suppression by denoiser projection, while overall trajectory errors remain small.

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