Efficient and robust persistent homology for measures

Mickael Buchet, Frederic Chazal, Steve Y. Oudot, Donald R. Sheehy

Research output: Chapter in Book/Report/Conference proceedingConference contributionResearchpeer-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
Title of host publicationProceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015
PublisherAssociation of Computing Machinery
Pages168-180
Number of pages13
Volume2015-January
EditionJanuary
Publication statusPublished - 1 Jan 2015
Externally publishedYes
Event26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015 - San Diego, United States
Duration: 4 Jan 20156 Jan 2015

Conference

Conference26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015
CountryUnited States
CitySan Diego
Period4/01/156/01/15

Fingerprint

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

ASJC Scopus subject areas

  • Software
  • Mathematics(all)

Cite this

Buchet, M., Chazal, F., Oudot, S. Y., & Sheehy, D. R. (2015). Efficient and robust persistent homology for measures. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015 (January ed., Vol. 2015-January, pp. 168-180). Association of Computing Machinery.

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

Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015. Vol. 2015-January January. ed. Association of Computing Machinery, 2015. p. 168-180.

Research output: Chapter in Book/Report/Conference proceedingConference contributionResearchpeer-review

Buchet, M, Chazal, F, Oudot, SY & Sheehy, DR 2015, Efficient and robust persistent homology for measures. in Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015. January edn, vol. 2015-January, Association of Computing Machinery, pp. 168-180, 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, United States, 4/01/15.
Buchet M, Chazal F, Oudot SY, Sheehy DR. Efficient and robust persistent homology for measures. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015. January ed. Vol. 2015-January. Association of Computing Machinery. 2015. p. 168-180
Buchet, Mickael ; Chazal, Frederic ; Oudot, Steve Y. ; Sheehy, Donald R. / Efficient and robust persistent homology for measures. Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015. Vol. 2015-January January. ed. Association of Computing Machinery, 2015. pp. 168-180
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