David Cohen-Steiner

NeurIPS Conference 2024 Conference Paper

Wasserstein convergence of Cech persistence diagrams for samplings of submanifolds

• Charles Arnal
• David Cohen-Steiner
• Vincent Divol

Cech Persistence diagrams (PDs) are topological descriptors routinely used to capture the geometry of complex datasets. They are commonly compared using the Wasserstein distances $\mathrm{OT}_p$; however, the extent to which PDs are stable with respect to these metrics remains poorly understood. We partially close this gap by focusing on the case where datasets are sampled on an $m$-dimensional submanifold of $\mathbb{R}^d$. Under this manifold hypothesis, we show that convergence with respect to the $\mathrm{OT}_p$ metric happens exactly when $p>m$. We also provide improvements upon the bottleneck stability theorem in this case and prove new laws of large numbers for the total $\alpha$-persistence of PDs. Finally, we show how these theoretical findings shed new light on the behavior of the feature maps on the space of PDs that are used in ML-oriented applications of Topological Data Analysis.

SODA Conference 2009 Conference Paper

Persistent homology for kernels, images, and cokernels

• David Cohen-Steiner
• Herbert Edelsbrunner
• John Harer
• Dmitriy Morozov

Motivated by the measurement of local homology and of functions on noisy domains, we extend the notion of persistent homology to sequences of kernels, images, and cokernels of maps induced by inclusions in a filtration of pairs of spaces. Specifically, we note that persistence in this context is well defined, we prove that the persistence diagrams are stable, and we explain how to compute them.

FOCS Conference 2007 Conference Paper

Inferring Local Homology from Sampled Stratified Spaces

• Paul Bendich
• David Cohen-Steiner
• Herbert Edelsbrunner
• John Harer
• Dmitriy Morozov

We study the reconstruction of a stratified space from a possibly noisy point sample. Specifically, we use the vineyard of the distance function restricted to a 1-parameter family of neighborhoods of a point to assess the local homology of the stratified space at that point. We prove the correctness of this assessment under the assumption of a sufficiently dense sample. We also give an algorithm that constructs the vineyard and makes the local assessment in time at most cubic in the size of the Delaunay triangulation of the point sample.

STOC Conference 2004 Conference Paper

Isotopic implicit surface meshing

• Jean-Daniel Boissonnat
• David Cohen-Steiner
• Gert Vegter