Abstract:We study denoising score matching (DSM) when the latent distribution is supported on a smooth embedded manifold $M \subset \mathbb{R}^D$. Under ambient Gaussian corruption, the tangent denoising target contains a singular normal-fiber noise channel whose variance diverges as $d/\sigma^2$ as $\sigma \to 0^+$. We show that conditioning on the nearest-point projection $\pi(X)$ canonically removes this singularity: the resulting conditional expectation is the unique $L^2$-optimal Rao-Blackwellized predictor of the tangent DSM target among all estimators depending only on the projected observation $\pi(X)$. We then compute the small-noise expansion of this canonical target and show that it equals the intrinsic Riemannian score up to an explicit order-$\sigma^2$ correction that decomposes into an intrinsic Tweedie term and an extrinsic curvature term involving the Weingarten and Ricci operators. In the flat case, the construction reduces exactly to ordinary lower-dimensional Gaussian DSM, while on $S^d$ the extrinsic correction simplifies to the scalar factor $(1-d/2)\nabla_M \log q$; this extrinsic $\sigma^2$ correction cancels identically on $S^2$, though the intrinsic Tweedie term remains.
| Comments: | 22 pages, 3 figures. SPIGM @ ICML 2026 |
| Subjects: | Machine Learning (stat.ML); Machine Learning (cs.LG) |
| Cite as: | arXiv:2605.25567 [stat.ML] |
| (or arXiv:2605.25567v1 [stat.ML] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25567 arXiv-issued DOI via DataCite (pending registration) |
From: Divit Rawal [view email]
[v1]
Mon, 25 May 2026 08:18:45 UTC (1,112 KB)
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