Combinatorial Invariants of Multidimensional Topological Network Data

TL;DR

This paper explores how combinatorial invariants of high-dimensional topological network data can be understood through linear matroids, providing new methods for analyzing point cloud data in computational topology.

Contribution

It introduces a matroid-theoretic framework to study combinatorial invariants of multidimensional topological network data, linking algebraic topology with matroid theory.

Findings

Matroid properties reflect meaningful invariants of topological complexes.

Derived estimates for summary statistics of random point cloud data.

Applicable to various sampling distributions in 2D and 3D.

Abstract

Modern applications of algebraic topology to point cloud data analysis have motivated active investigation of combinatorial clique complexes -- high-dimensional extensions of combinatorial graphs. We show that meaningful invariants of such spaces are reflected in the combinatorial properties of an associated family of linear matroids, and discuss matroid-theoretic approaches to several problems in computational topology. Our results allow us to derive estimates of the summary statistics of related constructs for random point cloud data, which we discuss for several sampling distributions in $\mathbb{R}^2$ and $\mathbb{R}^3$.