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This paper advances denoising score matching by deriving a Rao-Blackwellized estimator that corrects variance divergence on embedded manifolds under ambient noise. The authors provide a rigorous small-noise expansion revealing geometric biases related to curvature and embedding drift, while proving these vanish exactly on hyperspheres like $S^2$. This theoretical insight explains the empirical success of standard methods in Earth science applications without requiring complex intrinsic modeling for spherical data.
Rao-Blackwellized Score Matching on Manifolds
arXiv:2605.25567v3 Announce Type: replace-cross Abstract: We study denoising score matching (DSM) when data are drawn from an embedded manifold $M \subset \mathbb{R}^D$. We show that under ambient Gaussian corruption, the target has variance that diverges as the noise scale decreases and correct for it by regressing against the conditional expectation given the nearest point projection on the manifold: the $L^2$-optimal Rao-Blackwellized target. We then compute the small-noise expansion of this target and show that it recovers the true intrinsic Riemannian score to first order, with a second-order bias from a Tweedie term and two geometric terms dependent on how the manifold is embedded in ambient space: a curvature operator acting on the intrinsic score, and an additive drift generated by the spatial variation of the embedding's second fundamental form. On hyperspheres, we derive a simplified formula and show that both geometric terms vanish exactly on $S^2$, offering a theoretical explanation for why ambient DSM performs comparably to intrinsic methods on real Earth science spherical data in prior work.
Abstract:We study denoising score matching (DSM) when data are drawn from an embedded manifold $M \subset \mathbb{R}^D$. We show that under ambient Gaussian corruption, the target has variance that diverges as the noise scale decreases and correct for it by regressing against the conditional expectation given the nearest point projection on the manifold: the $L^2$-optimal Rao-Blackwellized target. We then compute the small-noise expansion of this target and show that it recovers the true intrinsic Riemannian score to first order, with a second-order bias from a Tweedie term and two geometric terms dependent on how the manifold is embedded in ambient space: a curvature operator acting on the intrinsic score, and an additive drift generated by the spatial variation of the embedding's second fundamental form. On hyperspheres, we derive a simplified formula and show that both geometric terms vanish exactly on $S^2$, offering a theoretical explanation for why ambient DSM performs comparably to intrinsic methods on real Earth science spherical data in prior work.
Machine Learning (stat.ML); Machine Learning (cs.LG)
arXiv:2605.25567 [stat.ML]
(or arXiv:2605.25567v3 [stat.ML] for this version)
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