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

arXiv:2606.06179v3 Announce Type: replace-cross 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. Recent work has shown that the score admits an orthogonal decomposition into gradient and solenoidal components in which only the gradient part matters for exact sampling. We apply this decomposition to the score error of trained models and study its quantitative consequences. Since only the gradient component of the error enters the marginal Fokker-Planck dynamics, a model can incur arbitrarily large $L^2$ score error while perfectly matching the target. Starting from this observation, we present three results. First, we derive an upper bound on the Kullback-Leibler divergence that depends only on the observable gradient component of the error. It tightens the standard Girsanov bound for generic score networks and identifies that bound's looseness as the cost of working with path-space rather than marginal-space dynamics. Second, we prove that no non-trivial function of the $L^2$ score error can uniformly lower bound any divergence between the learned and target distributions. Third, we give an estimator of the gradient component, computable from denoising score matching residuals via a dual Sobolev identity, which empirically correlates substantially better with sample quality than the full $L^2$ error.