The Loss Floor of Denoising Score Matching: Fisher Geometry from Schrödinger Bridges

What the math gives to ML

The paper identifies an exact irreducible loss component in denoising score matching: the conditional-score regression loss exceeds the marginal-score loss by the conditional variance of the target. This variance is the trace of a Fisher–Rao metric and, for Gaussian corruption, can be written using the posterior covariance of the clean sample given the noisy sample. The most actionable transfer is to subtract or normalize this analytically estimated floor when comparing models, selecting noise weights, or constructing curricula, preventing schedule-dependent noise from being mistaken for model quality. A second opportunity is information-flow weighting, using the identity between the floor and mutual-information decay to allocate training effort where the corruption channel is most informative.