The DTM-signature for a geometric comparison of metric-measure spaces from samples

TL;DR

This paper introduces the DTM-signature, a new measure for comparing metric-measure spaces that provides a practical statistical test for assessing their similarity based on samples.

Contribution

It proposes the DTM-signature as a novel geometric measure and develops a statistical test for space equality, with theoretical justifications and an algorithm.

Findings

The DTM-signature defines a pseudo-metric bounded by Gromov-Wasserstein distance.

A statistical test for space equality is developed and justified.

An algorithm for implementing the test is proposed.

Abstract

In this paper, we introduce the notion of DTM-signature, a measure on R + that can be associated to any metric-measure space. This signature is based on the distance to a measure (DTM) introduced by Chazal, Cohen-Steiner and M\'erigot. It leads to a pseudo-metric between metric-measure spaces, upper-bounded by the Gromov-Wasserstein distance. Under some geometric assumptions, we derive lower bounds for this pseudo-metric. Given two N-samples, we also build an asymptotic statistical test based on the DTM-signature, to reject the hypothesis of equality of the two underlying metric-measure spaces, up to a measure-preserving isometry. We give strong theoretical justifications for this test and propose an algorithm for its implementation.