Manifold Learning with Geodesic Minimal Spanning Trees

TL;DR

This paper introduces the GMST method, a simple geometric approach that estimates the intrinsic dimension and entropy of data on a manifold without reconstructing the manifold or density, demonstrated on face data.

Contribution

The paper presents a novel GMST approach that provides asymptotically consistent estimates of manifold dimension and entropy using minimal spanning trees based on geodesic distances.

Findings

GMST accurately estimates manifold dimension and entropy.

Method does not require manifold reconstruction or density estimation.

Effective on real-world face dataset.

Abstract

In the manifold learning problem one seeks to discover a smooth low dimensional surface, i.e., a manifold embedded in a higher dimensional linear vector space, based on a set of measured sample points on the surface. In this paper we consider the closely related problem of estimating the manifold's intrinsic dimension and the intrinsic entropy of the sample points. Specifically, we view the sample points as realizations of an unknown multivariate density supported on an unknown smooth manifold. We present a novel geometrical probability approach, called the geodesic-minimal-spanning-tree (GMST), to obtaining asymptotically consistent estimates of the manifold dimension and the R\'{e}nyi $\alpha$-entropy of the sample density on the manifold. The GMST approach is striking in its simplicity and does not require reconstructing the manifold or estimating the multivariate density of the samples. The GMST method simply constructs a minimal spanning tree (MST) sequence using a geodesic edge matrix and uses the overall lengths of the MSTs to simultaneously estimate manifold dimension and entropy. We illustrate the GMST approach for dimension and entropy estimation of a human face dataset.