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

Denoising score matching regresses onto a conditional score, while the generative dynamics require the marginal score. The two objectives share a population minimizer, but at fixed noisy state the conditional target is still random, and this inflates the training loss by a term no model can reduce. We show that this excess is exactly the trace of the Fisher–Rao metric of the conditional endpoint family, integrated along the diffusion, for any corruption kernel meeting mild regularity conditions. Deriving it from the Schr\"odinger bridge variational principle makes its origin precise, the first variation of the bridge functional gives the generative drift, the second gives the metric, so the information geometry observed in diffusion latent spaces is an intrinsic part of the training loss rather than a structure imposed on the model. For corruption diffusions the floor factorizes into a data-dependent information flow and a schedule-dependent weight, and for Gaussian corruption it evaluates to the mutual information the noisy state loses about the data. It is also measurable where, one extra backward pass recovers it from a trained network, and on CIFAR-10 it is $99.86\%$ of the reported loss on the noise range in standard use, large enough that two checkpoints of a single run are ranked backwards by a factor of $5.9$, an inversion that subtracting the floor repairs. The same decomposition applies to masked diffusion, where the floor is the entropy of the data and is likewise schedule-independent.