Tangent Space and Dimension Estimation with the Wasserstein Distance

TL;DR

This paper establishes rigorous bounds on the sample size needed for accurate tangent space and dimension estimation of manifolds using Local PCA, accommodating noise and non-uniform data distributions.

Contribution

It provides explicit bounds and a rigorous theoretical framework for tangent space and dimension estimation using Wasserstein distance and matrix concentration inequalities.

Findings

Explicit sample size bounds for manifold estimation

Robustness to noisy, non-uniform data distributions

Simultaneous estimation at multiple points

Abstract

Consider a set of points sampled independently near a smooth compact submanifold of Euclidean space. We provide mathematically rigorous bounds on the number of sample points required to estimate both the dimension and the tangent spaces of that manifold with high confidence. The algorithm for this estimation is Local PCA, a local version of principal component analysis. Our results accommodate for noisy non-uniform data distribution with the noise that may vary across the manifold, and allow simultaneous estimation at multiple points. Crucially, all of the constants appearing in our bound are explicitly described. The proof uses a matrix concentration inequality to estimate covariance matrices and a Wasserstein distance bound for quantifying nonlinearity of the underlying manifold and non-uniformity of the probability measure.