arXiv:2605.25567v1 Announce Type: cross
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.