Rao-Blackwellized Score Matching on Manifolds

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.