Persistence stability for geometric complexes

By Frédéric Chazal, Vin de Silva, Steve Y. Oudot. arXiv.org, 2012

346 citations recorded by the source corpus.

In this paper we study the properties of the homology of different geometric filtered complexes (such as Vietoris–Rips, Čech and witness complexes) built on top of totally bounded metric spaces. Using recent developments in the theory of topological persistence, we provide simple and natural proofs of the stability of the persistent homology of such complexes with respect to the Gromov–Hausdorff distance. We also exhibit a few noteworthy properties of the homology of the Rips and Čech complexes built on top of compact spaces.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/eae6ad102732385fc57dab6a5444a11f1c8d9d9c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. Chad Giusti
2. Frédéric Chazal
3. Omer Bobrowski
4. Primoz Skraba
5. Vin de Silva
6. W. Crawley‐Boevey
7. C. Peterson
8. Carina Curto

In Mathematical Physics

1. Omer Bobrowski
2. Vin de Silva
3. Primoz Skraba
4. Frédéric Chazal
5. W. Crawley‐Boevey
6. Chad Giusti
7. Carina Curto
8. Vladimir Itskov

In Radiology, Nuclear Medicine and Imaging

1. Primoz Skraba
2. W. Crawley‐Boevey
3. Omer Bobrowski
4. Chad Giusti
5. Vin de Silva
6. Frédéric Chazal

Citations in the corpus

• Clique topology reveals intrinsic geometric structure in neural correlations (2015, background): “Further work has broadened this result by developing more complete theoretical tools for the comparison of persistence structures, divorcing their stability from any underlying geometry [24, 25, 26].”
• Persistence Images: A Stable Vector Representation of Persistent Homology (2015, methodology): “…al., 2011; Turner et al., 2014)), and since these metrics reveal the stability of PDs under small perturbations of the data they summarize (Cohen-Steiner et al., 2007, 2010; Chazal et al., 2014), it is possible to perform a variety of ML techniques using PDs as a statistic for clustering data sets.”
• Wasserstein Stability for Persistence Diagrams (2020, neutral): “Additionally, stability results have been been shown for specific constructions, e.g. geometric complexes [11].”
• The observable structure of persistence modules (2014, background): “Then the Vietoris–Rips and intrinsic Čech filtered complexes on X have q-tame persistent homology [3].”
• Maximally Persistent Cycles in Random Geometric Complexes (2015, background): “approximation results for geometric complexes are multiplicative [18, 26, 48],”
• A Complete Characterization of the 1-Dimensional Intrinsic Cech Persistence Diagrams for Metric Graphs (2017, methodology): “This distance, introduced in [6], is defined as follows: dIC(G1,G2) := dB(Dg1ICG1 ,Dg1ICG2),”