Tight basis cycle representatives for persistent homology of large data sets

TL;DR

This paper introduces efficient algorithms for computing precise, tight boundaries around persistent homology features in large data sets, enabling detailed topological analysis in scientific applications.

Contribution

The authors develop new algorithms for localized, tight representatives of persistent homology features, improving accuracy and computational efficiency for large data analysis.

Findings

Identified unexpected chromatin loops in human genome data.

Detected statistically significant cosmic voids in galaxy distributions.

Attributed topological voids in proteins to biological interactions.

Abstract

Persistent homology (PH) is a popular tool for topological data analysis that has found applications across diverse areas of research. It provides a rigorous method to compute robust topological features in discrete experimental observations that often contain various sources of uncertainties. Although powerful in theory, PH suffers from high computation cost that precludes its application to large data sets. Additionally, most analyses using PH are limited to computing the existence of nontrivial features. Precise localization of these features is not generally attempted because, by definition, localized representations are not unique and because of even higher computation cost. For scientific applications, such a precise location is a sine qua non for determining functional significance. Here, we provide a strategy and algorithms to compute tight representative boundaries around nontrivial robust features in large data sets. To showcase the efficiency of our algorithms and the precision of computed boundaries, we analyze three data sets from different scientific fields. In the human genome, we found an unexpected effect on loops through chromosome 13 and the sex chromosomes, upon impairment of chromatin loop formation. In a distribution of galaxies in the universe, we found statistically significant voids. In protein homologs with significantly different topology, we found voids attributable to ligand-interaction, mutation, and differences between species.