# Tight basis cycle representatives for persistent homology of large data sets

arXiv: https://arxiv.org/abs/2206.02925 · HTML: https://mlsanity.com/paper/2206.02925/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 46 references that are in the map.
- First posted on arXiv 6 Jun 2022.

## Builds on (most cited first, 7 of 7)

- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Topology-Aware Surface Reconstruction for Point Clouds](https://mlsanity.com/paper/1811.12543/) (2018)
- [Minimal Cycle Representatives in Persistent Homology using Linear Programming: an Empirical Study with User's Guide](https://mlsanity.com/paper/2105.07025/) (2021)
- [Topological data analysis of contagion maps for examining spreading processes on networks](https://mlsanity.com/paper/1408.1168/) (2014)
- [Efficient algorithms for computing a minimal homology basis](https://mlsanity.com/paper/1801.06759/) (2018)
- [Homological Scaffold via Minimal Homology Bases](https://mlsanity.com/paper/2004.11606/) (2020)
- [Dory: Overcoming Barriers to Computing Persistent Homology](https://mlsanity.com/paper/2103.05608/) (2021)

## Built on by (most cited first, 1 of 1)

- [Scalable Topology-Preserving Graph Coarsening: Concepts and Algorithms](https://mlsanity.com/paper/2601.22943/) (2026)

## Abstract (arXiv)

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.