Estimation of Local Geometric Structure on Manifolds from Noisy Data

Abstract

A common observation in data-driven applications is that high-dimensional data have a low intrinsic dimension, at least locally. In this work, we consider the problem of point estimation for manifold-valued data. Namely, given a finite set of noisy samples of $\mathcal{M}$, a $d$ dimensional submanifold of ${\mathbb{R}}^D$, and a point $r$ near the manifold we aim to project $r$ onto the manifold. Assuming that the data was sampled uniformly from a tubular neighborhood of a $k$-times smooth boundaryless and compact manifold, we present an algorithm that takes $r$ from this neighborhood and outputs ${\hat{p}}_n\in {\mathbb{R}}^D$, and $\hat{T_{{\hat{p}}_n}\mathcal{M}}$ an element in the Grassmannian $Gr(d,D)$. We prove that as the number of samples $n\to \infty$, the point ${\hat{p}}_n$ converges to $\mathbf{p}\in \mathcal{M}$, the projection of $r$ onto $\mathcal{M}$, and $\hat{T_{{\hat{p}}_n}\mathcal{M}}$ converges to $T_{\mathbf{p}}\mathcal{M}$ (the tangent space at that point) with high probability. Furthermore, we show that ${\hat{p}}_n$ approaches the manifold with an asymptotic rate of $n^{-\frac{k}{2k+d}}$, and that ${\hat{p}}_n,\hat{T_{{\hat{p}}_n}\mathcal{M}}$ approach $\mathbf{p}$ and $T_{\mathbf{p}}\mathcal{M}$ correspondingly with asymptotic rates of $n^{-\frac{k-1}{2k+d}}$.