The Manifold Density Function: An Intrinsic Method for the Validation of Manifold Learning

TL;DR

This paper introduces the manifold density function, an intrinsic validation method for manifold learning that generalizes Ripley's K-function to broad classes of Riemannian manifolds, with proven convergence and robustness.

Contribution

It extends the manifold density function to general two-manifolds and hypersurfaces, providing a new intrinsic validation tool for manifold learning algorithms.

Findings

Validates manifold learning with intrinsic measures

Generalizes to broad classes of Riemannian manifolds

Proves convergence and robustness properties

Abstract

We introduce the manifold density function, which is an intrinsic method to validate manifold learning techniques. Our approach adapts and extends Ripley's $K$-function, and categorizes in an unsupervised setting the extent to which an output of a manifold learning algorithm captures the structure of a latent manifold. Our manifold density function generalizes to broad classes of Riemannian manifolds. In particular, we extend the manifold density function to general two-manifolds using the Gauss-Bonnet theorem, and demonstrate that the manifold density function for hypersurfaces is well approximated using the first Laplacian eigenvalue. We prove desirable convergence and robustness properties.