Estimation of Riemannian Quantities from Noisy Data via Density Derivatives (opens in new tab)

We study the recovery of geometric structure from data generated by convolving the uniform measure on a smooth compact submanifold $M\subset\mathbb{R}^D$ with ambient Gaussian noise. Our main result is that several fundamental Riemannian quantities of $M$, including tangent spaces, the intrinsic dimension, and the second fundamental form, are identifiable from derivatives of the noisy density. We first derive uniform small-noise expansions of t...