Diffusion Models Observe Only Gradients: A Geometric Perspective on Score Matching Errors

• URL: http://arxiv.org/abs/2606.06179v1
• Date: Thu, 04 Jun 2026 13:53:38 GMT
• Title: Diffusion Models Observe Only Gradients: A Geometric Perspective on Score Matching Errors
• Abstract summary: We show the $L2$ score error is not the right intrinsic measure of marginal distributional quality.<n>A learned diffusion model can incur large $L2$ score error while perfectly matching the target distribution.
• Score: 27.894241484593735
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Score-based diffusion models are typically trained by minimizing the $L^2$ score matching error, and standard theoretical analyses rely on this quantity to bound the sampling discrepancy between the learned and target distributions. We show the $L^2$ score error is not the right intrinsic measure of marginal distributional quality: a learned diffusion model can incur arbitrarily large $L^2$ score error while perfectly matching the target distribution. By decomposing score errors into a gradient and a solenoidal component (a Helmholtz-Hodge decomposition), we identify the geometric reason behind this: only the gradient component enters the marginal Fokker-Planck dynamics, while the solenoidal component is structurally invisible. We make this precise in three results. First, building on the corrected geometry, we prove an impossibility result: no monotone function of the $L^2$ score error can uniformly lower bound any divergence between the learned and target distributions. Second, we derive an upper bound on the Kullback-Leibler divergence that depends only on the observable gradient component of the error, tightening the standard Girsanov bound and identifying its looseness as the cost of operating on path-space rather than marginal-space dynamics. Third, we give a tractable estimator of the gradient component via a dual Sobolev identity, which is shown to empirically correlate substantially better with sample quality than the full $L^2$ error.

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