Convergence of the Reach for a Sequence of Gaussian-Embedded Manifolds

TL;DR

This paper investigates the asymptotic behavior of the reach, a measure of manifold curvature and topology, for a sequence of Gaussian-embedded manifolds, providing convergence and fluctuation results.

Contribution

It develops the first limit theory for the reach of Gaussian-embedded manifolds, including almost sure convergence and fluctuation analysis.

Findings

Global reach converges to a known constant in Gaussian process theory.

Established almost sure convergence of the global reach.

Provided fluctuation theory for both global and local reach.

Abstract

Motivated by questions of manifold learning, we study a sequence of random manifolds, generated by embedding a fixed, compact manifold $M$ into Euclidean spheres of increasing dimension via a sequence of Gaussian mappings. One of the fundamental smoothness parameters of manifold learning theorems is the reach, or critical radius, of $M$. Roughly speaking, the reach is a measure of a manifold's departure from convexity, which incorporates both local curvature and global topology. This paper develops limit theory for the reach of a family of random, Gaussian-embedded, manifolds, establishing both almost sure convergence for the global reach, and a fluctuation theory for both it and its local version. The global reach converges to a constant well known both in the reproducing kernel Hilbert space theory of Gaussian processes, as well as in their extremal theory.