Estimation of Riemannian Quantities from Noisy Data via Density Derivatives

TL;DR

This paper demonstrates how to recover key Riemannian geometric quantities of a manifold from noisy data by analyzing derivatives of the data density, with theoretical guarantees and practical estimators.

Contribution

It introduces a method to estimate tangent spaces, intrinsic dimension, and second fundamental form from noisy data using density derivatives, with proven error bounds.

Findings

Tangent spaces can be recovered from the density Hessian with $O(\sigma^2)$ error.

Intrinsic dimension can be estimated consistently from noisy data.

Constructed estimators for the second fundamental form achieve specific error rates depending on the manifold.

Abstract

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 the data density and its derivatives in a tubular neighborhood of $M$. These expansions show that, at the population level, tangent spaces can be recovered from the density Hessian with $O(\sigma^2)$ error, while the intrinsic dimension can be estimated consistently. We further construct estimators for the second fundamental form from density derivatives, obtaining $O(d(y,M)+\sigma)$ and $O(d(y,M)+\sigma^2)$ errors for hypersurfaces and submanifolds with arbitrary codimension. At the sample level, we estimate the density and its derivatives by kernel methods in the ambient space and plug them into the population constructions, yielding uniform nonparametric rates in the ambient dimension. Finally, we show that these density-based constructions admit a geometric interpretation through density-induced ambient metrics, linking the geometry of $M$ to ambient geodesic structure.