Persistence stability for geometric complexes

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.

💡 Research Summary

The paper investigates the homology of several geometric filtered complexes—Vietoris‑Rips, Čech, and witness complexes—constructed on precompact metric spaces. By leveraging recent advances in persistent homology, the authors give concise and natural proofs that the persistent homology of these complexes is stable with respect to the Gromov–Hausdorff distance (dGH).

First, the authors define the ε‑filtered families Rε(X), Cε(X), and Wε(L,W) for a precompact space X, a landmark set L, and a witness set W. They then show that if two spaces X and Y satisfy dGH(X,Y) ≤ δ, there exist inclusion maps Rε+δ(X) → Rε(Y) and Cε+δ(X) → Cε(Y) that form a 2δ‑interleaving of the corresponding persistence modules. Consequently, the bottleneck distance between the persistence diagrams of the two spaces is bounded by 2δ, establishing a 2‑Lipschitz continuity of persistent homology under dGH.

For compact spaces, the authors prove a “stabilization” phenomenon: once ε exceeds the diameter of X, both the Rips and Čech complexes become homotopy equivalent to X itself, causing higher‑dimensional homology groups to vanish. They introduce a stabilization index that records the ε‑value at which the barcode ceases to change. Moreover, they demonstrate that for sufficiently large ε the Rips complex approximates the Čech complex, linking the two constructions via the nerve theorem.

The witness complex analysis shows that if the landmark set L is δ‑dense in X, then for any ε > δ the witness complex Wε(L,W) interleaves with the Rips complex Rε+δ(X). This yields the same dGH‑stability result for witness complexes and provides a theoretical justification for using subsampled data to approximate the persistent homology of the full space.

The paper also discusses subtle differences between Rips and Čech complexes. In dimension one, the Rips complex can generate spurious cycles that disappear as ε grows, whereas the Čech complex, by virtue of the nerve theorem, accurately reflects the underlying topology for sufficiently small ε. These observations guide practitioners in choosing the appropriate complex for a given application.

Finally, the authors outline future directions: extending stability results to non‑compact spaces, developing computationally efficient approximations, and exploring multi‑scale interactions between different complexes. By simplifying earlier, more intricate proofs and clarifying the relationship between Gromov–Hausdorff distance and persistent homology, the paper provides a solid theoretical foundation for topological data analysis on geometric complexes.