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

• URL: http://arxiv.org/abs/2607.08757v1
• Date: Thu, 09 Jul 2026 17:55:52 GMT
• Title: Score Accuracy Along the Forward Diffusion Does Not Certify Numerical Stability in Diffusion Sampling
• Abstract summary: We show that small forward-marginal error does not guarantee numerical stability.<n>We construct a single smooth score field with arbitrarily small forward-marginal $L2$ error.<n>Experiments with a small fixed DiT-style network show large growth along rare numerical trajectories.
• Score: 1.218340575383456
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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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