Simplexwise Distance Distributions for finite spaces with metrics and measures

TL;DR

This paper introduces Simplexwise Distance Distributions (SDDs), a novel invariant for finite metric and metric-measure spaces, enabling precise classification and comparison of complex geometric structures.

Contribution

It develops SDDs as a new invariant for finite metric spaces, with polynomial-time computability and Lipschitz continuity, improving space classification methods.

Findings

SDDs classify all known non-equivalent finite metric spaces.

SDDs can be computed exactly with polynomial complexity.

SDDs are Lipschitz continuous metrics on the space of invariants.

Abstract

A finite set of unlabelled points in Euclidean space is the simplest representation of many real objects from mineral rocks to sculptures. Since most solid objects are rigid, their natural equivalence is rigid motion or isometry maintaining all inter-point distances. More generally, any finite metric space is an example of a metric-measure space that has a probability measure and a metric satisfying all axioms. This paper develops Simplexwise Distance Distributions (SDDs) for any finite metric spaces and metric-measures spaces. These SDDs classify all known non-equivalent spaces that were impossible to distinguish by simpler invariants. We define metrics on SDDs that are Lipschitz continuous and allow exact computations whose parametrised complexities are polynomial in the number of given points.