Persistence stability for geometric complexes (1207.3885v3)

Abstract: In this paper we study the properties of the homology of different geometric filtered complexes (such as Vietoris-Rips, Cech and witness complexes) built on top of precompact 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 Cech complexes built on top of compact spaces.

• The paper introduces a novel ε-simplicial multivalued map framework that enables ε-interleavings for persistent homology stability.
• It establishes that the bottleneck distance between persistence diagrams is bounded by twice the Gromov-Hausdorff distance.
• The study extends stability results to totally bounded, path metric, and δ-hyperbolic spaces, providing actionable insights for computational topology.

Persistence Stability for Geometric Complexes

The paper "Persistence stability for geometric complexes" by Frederic Chazal, Vin de Silva, and Steve Oudot provides a thorough investigation into the properties of the homology of various geometric filtered complexes, such as Vietoris-Rips, Cech, and witness complexes, in the context of totally bounded metric spaces. It employs modern advancements in the theory of topological persistence to deliver proofs for the stability of persistent homology concerning the Gromov-Hausdorff distance. The paper further explores the homological properties of Rips and Cech complexes over compact spaces.

The task of deducing topological attributes of metric spaces from finite approximations has gained traction in computational topology. The paper addresses this challenge by focusing on metric spaces where a simplicial complex is constructed from a finite vertex set to approximate the homology or homotopy type of an unknown space. The paper extensively discusses the utility of the Rips complex due to its computational simplicity and effective approximation characteristics.

J.C. Hausmann and J. Latschev’s works are foundational in this space, showcasing Rips complexes’ potential to capture homotopy equivalence under specific conditions. However, selecting a suitable scale parameter $\alpha$ remains daunting due to its dependency on the intrinsic geometry of the space in question, complicating its application in real-world data.

Topological persistence offers a promising resolution by encoding homological data across a spectrum of scale parameters into a persistence diagram. The diagram provides insights into the relationship between a chosen scale and the homology of the respective Vietoris-Rips complex. The stability of persistence diagrams, initially established for finite metric spaces, is foundational, with the paper extending this to totally bounded environments, introducing Dowker complexes for further analysis.

Key contributions include:

• Development of a conceptual framework extending simplicial maps to "ε-simplicial multivalued maps," facilitating canonical "ε-interleavings" in persistent homology modules.
• Establishing "ε-interleaving" of persistent homology for "ε-close" metric spaces, enhancing the analytical toolset for filtered geometric complexes.
• Confirming the tameness of persistent homology modules for totally bounded spaces, generalizing known stability bounds to these.

The paper demonstrates that for persistent modules in totally bounded spaces, the inequality $d_b(dgm(Rips(X)), dgm(Rips(Y))) \leq 2d_{GH}(X,Y)$, where $d_{GH}$ signifies Gromov-Hausdorff distance and $d_b$ represents the bottleneck distance, holds robustly.

Furthermore, Section 5.2 identifies notable peculiarities in the homology of Rips and Cech complexes, highlighting situations where homology groups display infinite dimensionality at particular parameter values, contrasting with their "well-behaved" persistent counterparts over totally bounded spaces.

The paper concludes with insights into persistent homology for path metric and δ-hyperbolic metric spaces, indicating no emergent 1-cycles beyond positive diameter for path metrics, and constraining new 2-cycles within 2δ for δ-hyperbolics.

This research both corroborates and amplifies the stability theorems for persistent homology, opening pathways for further theoretical exploration and practical application. Future work could focus on refining interleaving frameworks and expanding stability principles across broader classes of metric and dissimilarity spaces, enriching the landscape of geometric data analysis.