Approximating Loops in a Shortest Homology Basis from Point Data

TL;DR

This paper introduces an algorithm to approximate the shortest basis of the first homology group of a manifold from point data, advancing topological data analysis by providing a practical method for topological feature extraction.

Contribution

It presents the first algorithm to approximate the shortest homology basis from point samples of a manifold, and offers a polynomial-time method for computing shortest bases in weighted simplicial complexes.

Findings

Algorithm effectively approximates shortest homology bases from point data.

Provides polynomial-time algorithm for shortest basis computation in weighted complexes.

Advances topological data analysis by enabling practical topological feature extraction.

Abstract

Inference of topological and geometric attributes of a hidden manifold from its point data is a fundamental problem arising in many scientific studies and engineering applications. In this paper we present an algorithm to compute a set of loops from a point data that presumably sample a smooth manifold $M\subset \mathbb{R}^d$. These loops approximate a {\em shortest} basis of the one dimensional homology group $H_1(M)$ over coefficients in finite field $\mathbb{Z}_2$. Previous results addressed the issue of computing the rank of the homology groups from point data, but there is no result on approximating the shortest basis of a manifold from its point sample. In arriving our result, we also present a polynomial time algorithm for computing a shortest basis of $H_1(K)$ for any finite {\em simplicial complex} $K$ whose edges have non-negative weights.