# Stable Volumes for Persistent Homology

arXiv: https://arxiv.org/abs/2109.11711 · HTML: https://mlsanity.com/paper/2109.11711/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Sep 2021.

## Built on by (most cited first, 2 of 2)

- [Persistent homology-based descriptor for machine-learning potential of amorphous structures](https://mlsanity.com/paper/2206.13727/) (2022)
- [Geometric Localization of Homology Cycles](https://mlsanity.com/paper/2406.03183/) (2024)

## Abstract (arXiv)

This paper proposes a stable volume and a stable volume variant, referred to as a stable sub-volume, for more reliable data analysis using persistent homology. In prior research, an optimal cycle and similar ideas have been proposed to identify the homological structure corresponding to each birth-death pair in a persistence diagram. While this is helpful for data analysis using persistent homology, the results are sensitive to noise. In this paper, stable volumes and stable sub-volumes are proposed to solve this problem. For a special case, we prove that a stable volume is the robust part of an optimal volume against noise. We implemented stable volumes and sub-volumes on HomCloud, a data analysis software package based on persistent homology, and show examples of stable volumes and sub-volumes.