### 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 language | English |
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Title of host publication | Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015 |

Publisher | Association of Computing Machinery |

Pages | 168-180 |

Number of pages | 13 |

Volume | 2015-January |

Edition | January |

Publication status | Published - 1 Jan 2015 |

Externally published | Yes |

Event | 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015 - San Diego, United States Duration: 4 Jan 2015 → 6 Jan 2015 |

### Conference

Conference | 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015 |
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Country | United States |

City | San Diego |

Period | 4/01/15 → 6/01/15 |

### Fingerprint

### ASJC Scopus subject areas

- Software
- Mathematics(all)

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Research › peer-review

*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.

}

TY - GEN

T1 - Efficient and robust persistent homology for measures

AU - Buchet, Mickael

AU - Chazal, Frederic

AU - Oudot, Steve Y.

AU - Sheehy, Donald R.

PY - 2015/1/1

Y1 - 2015/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84938279798&partnerID=8YFLogxK

M3 - Conference contribution

VL - 2015-January

SP - 168

EP - 180

BT - Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015

PB - Association of Computing Machinery

ER -