We study denoising score matching (DSM) when the latent distribution is supported on a smooth embedded manifold
M⊂RD. Under ambient Gaussian corruption, the tangent denoising target contains a singular normal-fiber noise channel whose variance diverges as
d/σ2 as
σ→0+. We show that conditioning on the nearest-point projection
π(X) canonically removes this singularity: the resulting conditional expectation is the unique
L2-optimal Rao-Blackwellized predictor of the tangent DSM target among all estimators depending only on the projected observation
π(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-
σ2 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
Sd the extrinsic correction simplifies to the scalar factor
(1−d/2)∇Mlogq; this extrinsic
σ2 correction cancels identically on
S2, though the intrinsic Tweedie term remains.