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

arXiv · AI, language, vision and robotics · article · Aug 24, 2026 · UTC

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-varianc

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