Computing persistent homology by spanning trees and critical simplices

TL;DR

This paper introduces a novel method for computing persistent homology using spanning trees and critical simplices, enabling efficient calculation of topological features in high-dimensional data.

Contribution

It develops a new approach based on critical simplices and spanning trees to compute persistent homology more efficiently than existing methods.

Findings

The method achieves theoretical minimum critical simplices.

It quickly computes Betti numbers and cavity compositions.

It outperforms other methods in effectiveness and feasibility.

Abstract

Topological data analysis can extract effective information from higher-dimensional data. Its mathematical basis is persistent homology. The persistent homology can calculate topological features at different spatiotemporal scales of the dataset; that is, establishing the integrated taxonomic relation among points, lines and simplices. Here, the simplicial network composed of all-order simplices in a simplicial complex is essential. Because the sequence of nested simplicial subnetworks can be regarded as a discrete Morse function from the simplicial network to real values, a method based on the concept of critical simplices can be developed by searching all-order spanning trees. Employing this new method, not only the Morse function values with the theoretical minimum number of critical simplices can be obtained, but also the Betti numbers and composition of all-order cavities in the simplicial network can be calculated quickly. Finally, this method is used to analyze some examples and compared with other methods, showing its effectiveness and feasibility.