Tight basis cycle representatives for persistent homology of large data sets
It addresses the problem of enabling precise topological feature localization in large datasets for scientific applications, representing an incremental improvement over existing methods.
The paper tackles the high computational cost and lack of precise localization in persistent homology for large datasets, providing algorithms that compute tight representative boundaries around topological features, as demonstrated by analyzing human genome, galaxy distribution, and protein homolog data to reveal specific effects like loops in chromosomes and statistically significant voids.
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.