Efficient and robust persistent homology for measures

Mickaël Buchet, Frédéric Chazal, Steve Y. Oudot, Donald R. Sheehy

Research output: Contribution to journalArticleResearchpeer-review

Abstract

A new paradigm for point cloud data analysis has emerged recently, where point clouds are no longer treated as mere compact sets but rather as empirical measures. A notion of distance to such measures has been defined and shown to be stable with respect to perturbations of the measure. This distance can easily be computed pointwise in the case of a point cloud, but its sublevel-sets, which carry the geometric information about the measure, remain hard to compute or approximate. This makes it challenging to adapt many powerful techniques based on the Euclidean distance to a point cloud to the more general setting of the distance to a measure on a metric space. We propose an efficient and reliable scheme to approximate the topological structure of the family of sublevel-sets of the distance to a measure. We obtain an algorithm for approximating the persistent homology of the distance to an empirical measure that works in arbitrary metric spaces. Precise quality and complexity guarantees are given with a discussion on the behavior of our approach in practice.

Original languageEnglish
Pages (from-to)70-96
Number of pages27
JournalComputational Geometry: Theory and Applications
Volume58
DOIs
Publication statusPublished - 1 Oct 2017
Externally publishedYes

Fingerprint

Point Cloud
Homology
Empirical Measures
Metric space
Topological Structure
Euclidean Distance
Compact Set
Data analysis
Paradigm
Perturbation
Arbitrary

Keywords

  • Distance to a measure
  • Persistent homology
  • Power distance
  • Sparse rips filtration
  • Topological data analysis

ASJC Scopus subject areas

  • Computer Science Applications
  • Geometry and Topology
  • Control and Optimization
  • Computational Theory and Mathematics
  • Computational Mathematics

Cite this

Efficient and robust persistent homology for measures. / Buchet, Mickaël; Chazal, Frédéric; Oudot, Steve Y.; Sheehy, Donald R.

In: Computational Geometry: Theory and Applications, Vol. 58, 01.10.2017, p. 70-96.

Research output: Contribution to journalArticleResearchpeer-review

Buchet, Mickaël ; Chazal, Frédéric ; Oudot, Steve Y. ; Sheehy, Donald R. / Efficient and robust persistent homology for measures. In: Computational Geometry: Theory and Applications. 2017 ; Vol. 58. pp. 70-96.
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