Estimation of Local Geometric Structure on Manifolds from Noisy Data

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 M, a d dimensional submanifold of 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 p̂_n ∈ R^D, and (Formula presented).an element in the Grassmannian Gr(d,D). We prove that as the number of samples n → ∞, the point p̂_n converges to p ∈ M, the projection of r onto M, and (Formula presented) converges to T_pM (the tangent space at that point) with high probability. Furthermore, we show that p̂_n approaches the manifold with k an asymptotic rate of (Formula presented), and that (Formula presented).p and T_pM correspondingly k−1 with asymptotic rates of (Formula presented).