2025/03/10 by Aizenbud, Yariv, Sober, Barak · 1 citation
#FOS: Mathematics #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.2503.07220
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 ℝD, 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 pn∈ ℝD, and \widehatT pnM an element in the Grassmannian Gr(d, D). We prove that as the number of samples n→∞, the point pn converges to p∈ M, the projection of r onto M, and \widehatT pnM converges to TpM (the tangent space at that point) with high probability. Furthermore, we show that pn approaches the manifold with an asymptotic rate of n-(k)/(2k + d), and that pn, \widehatT pnM approach p and TpM correspondingly with asymptotic rates of n-(k-1)/(2k + d).