Tangent space estimation for smooth embeddings of Riemannian manifolds

TL;DR

This paper provides a theoretical analysis of how local sampling conditions affect tangent space estimation accuracy for smooth manifold embeddings, incorporating curvature and higher-order effects.

Contribution

It introduces a detailed theoretical framework for tangent space estimation considering second-order manifold properties and random sampling conditions.

Findings

Sampling width and density critically influence tangent space estimation accuracy.

Curvature and higher-order terms significantly affect local sampling requirements.

Numerical simulations confirm the theoretical bounds and insights.

Abstract

Numerous dimensionality reduction problems in data analysis involve the recovery of low-dimensional models or the learning of manifolds underlying sets of data. Many manifold learning methods require the estimation of the tangent space of the manifold at a point from locally available data samples. Local sampling conditions such as (i) the size of the neighborhood (sampling width) and (ii) the number of samples in the neighborhood (sampling density) affect the performance of learning algorithms. In this work, we propose a theoretical analysis of local sampling conditions for the estimation of the tangent space at a point P lying on a m-dimensional Riemannian manifold S in R^n. Assuming a smooth embedding of S in R^n, we estimate the tangent space T_P S by performing a Principal Component Analysis (PCA) on points sampled from the neighborhood of P on S. Our analysis explicitly takes into account the second order properties of the manifold at P, namely the principal curvatures as well as the higher order terms. We consider a random sampling framework and leverage recent results from random matrix theory to derive conditions on the sampling width and the local sampling density for an accurate estimation of tangent subspaces. We measure the estimation accuracy by the angle between the estimated tangent space and the true tangent space T_P S and we give conditions for this angle to be bounded with high probability. In particular, we observe that the local sampling conditions are highly dependent on the correlation between the components in the second-order local approximation of the manifold. We finally provide numerical simulations to validate our theoretical findings.