Diffusion Models Observe Only Gradients: A Geometric Perspective on Score Matching Errors
For researchers and practitioners training score-based diffusion models, this work identifies a fundamental flaw in the standard L2 score error metric and offers a better alternative for monitoring and improving sample quality.
The paper shows that the L2 score matching error is not a reliable measure of diffusion model quality because only its gradient component affects the marginal distribution; the solenoidal component is invisible. They prove an impossibility result, derive a tighter KL bound using only the gradient component, and provide a tractable estimator that correlates better with sample quality.
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.