From Varadhan's Limit to Eigenmaps: A Guide to the Geometric Analysis behind Manifold Learning

TL;DR

This paper reviews the development of heat kernel and eigenfunction theory on Riemannian manifolds and their applications to high-dimensional data analysis through spectral embeddings and manifold learning techniques.

Contribution

It provides a comprehensive overview connecting geometric analysis of manifolds with modern spectral methods for data dimension reduction, including recent theoretical advances.

Findings

Boundaries of heat kernels on various manifolds

Convergence notions for Riemannian manifolds

Uniform control of spectral embeddings on key classes

Abstract

We present an overview of the history of the heat kernel and eigenfunctions on Riemannian manifolds and how the theory has lead to modern methods of analyzing high dimensional data via eigenmaps and other spectral embeddings. We begin with Varadhan's Theorem relating the heat kernel to the distance function on a Riemannian manifold. We then review various theorems which bound the heat kernel on classes of Riemannian manifolds. Next we turn to eigenfunctions, the Sturm-Liouville Decomposition of the heat kernel using eigenfunctions, and various theorems which bound eigenfunctions on classes of Riemannian manifolds. We review various notions of convergence of Riemannian manifolds and which classes of Riemannian manifolds are compact with respect to which notions of convergence. We then present B\'erard-Besson-Gallot's heat kernel embeddings of Riemannian manifolds and the truncation of those embeddings. Finally we turn to Applications of Spectral embeddings to the Dimension Reduction of data sets lying in high dimensional spaces reviewing, in particular, the work of Belkin-Niyogi and Coifman-Lafon. We also review the Spectral Theory of Graphs and the work of Dodziuk and Chung and others. We close with recent theorems of Portegies and of the first author controlling truncated spectral embeddings uniformly on key classes of Riemannian manifolds. Throughout we provide many explicitly computed examples and graphics and attempt to provide as complete a set of references as possible. We hope that this article is accessible to both pure and applied mathematicians working in Geometric Analysis and their doctoral students.