Similarity Between Points in Metric Measure Spaces

TL;DR

This paper introduces a computationally feasible alternative to the Gromov-Hausdorff-Prokhorov distance for measuring similarity between points in metric measure spaces, focusing on neighborhood distributions.

Contribution

The paper proposes the radial distribution distance as a tractable, coarser similarity measure for points in metric measure spaces, replacing the NP-hard Gromov-Hausdorff-Prokhorov distance.

Findings

Radial distribution distance is easier to compute.

It provides a coarser but practical similarity measure.

The approach is applicable to various metric measure spaces.

Abstract

This paper is about similarity between objects that can be represented as points in metric measure spaces. A metric measure space is a metric space that is also equipped with a measure. For example, a network with distances between its nodes and weights assigned to its nodes is a metric measure space. Given points x and y in different metric measure spaces or in the same space, how similar are they? A well known approach is to consider x and y similar if their neighborhoods are similar. For metric measure spaces, similarity between neighborhoods is well captured by the Gromov-Hausdorff-Prokhorov distance, but it is NP-hard to compute this distance even in quite simple cases. We propose a tractable alternative: the radial distribution distance between the neighborhoods of x and y. The similarity measure based on the radial distribution distance is coarser than the similarity based on the Gromov-Hausdorff-Prokhorov distance but much easier to compute.