The Loss Floor of Denoising Score Matching: Fisher Geometry from Schr\"odinger Bridges

Abstract

Denoising score matching trains diffusion models by regressing onto a conditional score, although generation ultimately requires the marginal score. The two objectives share the same population minimizer, but the conditional target remains random at fixed noisy state and introduces an irreducible excess in the training loss. We isolate this excess and show that, for a general corruption kernel under mild regularity assumptions, it is exactly the trace of the Fisher--Rao metric of the conditional endpoint family, integrated along the diffusion trajectory. This gives an exact conditional-variance decomposition of the denoising objective and identifies the information geometry observed in diffusion latent spaces as an intrinsic component of the training loss. We derive the result from a Schr"odinger bridge variational principle, in which the ideal objective arises as excess path-space relative entropy. For corruption diffusions, the Fisher term is proportional to the rate at which the noisy state loses mutual information about the clean data, separating the loss floor into an information flow determined by the data and a weight determined by the corruption schedule and objective. In the Gaussian case, this yields a closed form for the floor, recovers reparametrization invariance of the continuous-time objective, and relates its high-SNR divergence to the information dimension of the data. Finally, we show that raw losses obtained with different noise ranges or weightings need not rank models consistently because they contain different additive floors, and contrast the second-order geometry seen by training with the third-order conditional statistics entering numerical sampling error.