Greedy Matroid Algorithm And Computational Persistent Homology

TL;DR

This paper introduces a greedy matroid algorithm for selecting an optimal basis in computational topology, specifically for estimating homology from samples, and explores its probabilistic properties.

Contribution

It presents a novel greedy matroid algorithm for basis selection in homology estimation and analyzes its relationship with sampling probabilities.

Findings

The algorithm efficiently finds an optimal basis for homology images.

The method provides a statistical approach to homology calculation from samples.

Analysis of sampling probabilities enhances understanding of the algorithm's effectiveness.

Abstract

An important problem in computational topology is to calculate the homology of a space from samples. In this work, we develop a statistical approach to this problem by calculating the expected rank of an induced map on homology from a sub-sample to the full space. We develop a greedy matroid algorithm for finding an optimal basis for the image of the induced map, and investigate the relationship between this algorithm and the probability of sampling vectors in the image of the induced map.