Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Persistent Cost of Lipschitz Maps

Francisco J. Gozzi, Manuela A. Cerdeiro, Pablo E. Riera

TL;DR

This work addresses how 1-Lipschitz maps between finite metric spaces affect persistent homology; it develops a geometric notion, the persistent cost $C(f)$, defined via the interleaving distances of kernel and cokernel persistence modules. The authors derive a metric upper bound $C(f) \le \operatorname{dist}(f) + 2\, d_H(f(X),Y)$ and establish stability of $C(f)$ under a Gromov–Hausdorff-type distance on maps, with $|C(f)-C(g)| \le 2\, d_{GH}(f,g)$. They also provide a self-contained treatment of stability for kernel, cokernel, and image modules and introduce a map-distance framework that refines classical GH stability. A metric example shows that algebraically optimal interleavings may not be realizable by any $1$-Lipschitz map, highlighting the distinction between abstract persistence isomorphisms and geometric realizability. Overall, the paper delivers concrete tools to compare datasets under Lipschitz maps and connects algebraic persistence with geometric distortions, with implications for robust data analysis under mappings $f:X\to Y$.

Abstract

A 1-Lipschitz map between compact metric spaces $f:X\to Y$ induces a homomorphism on the level of persistent d-homology of the associated Vietoris-Rips simplicial filtrations. This homomorphism naturally bounds the Bottleneck distance between said modules in terms of the maximum size of its kernel and cokernel persistence modules, a stable invariant which we refer to as the persistent cost associated to $f$. We provide metric upper bounds on the persistent cost of a 1-Lipschitz map and revisit its stability.

Persistent Cost of Lipschitz Maps

TL;DR

This work addresses how 1-Lipschitz maps between finite metric spaces affect persistent homology; it develops a geometric notion, the persistent cost , defined via the interleaving distances of kernel and cokernel persistence modules. The authors derive a metric upper bound and establish stability of under a Gromov–Hausdorff-type distance on maps, with . They also provide a self-contained treatment of stability for kernel, cokernel, and image modules and introduce a map-distance framework that refines classical GH stability. A metric example shows that algebraically optimal interleavings may not be realizable by any -Lipschitz map, highlighting the distinction between abstract persistence isomorphisms and geometric realizability. Overall, the paper delivers concrete tools to compare datasets under Lipschitz maps and connects algebraic persistence with geometric distortions, with implications for robust data analysis under mappings .

Abstract

A 1-Lipschitz map between compact metric spaces induces a homomorphism on the level of persistent d-homology of the associated Vietoris-Rips simplicial filtrations. This homomorphism naturally bounds the Bottleneck distance between said modules in terms of the maximum size of its kernel and cokernel persistence modules, a stable invariant which we refer to as the persistent cost associated to . We provide metric upper bounds on the persistent cost of a 1-Lipschitz map and revisit its stability.

Paper Structure

This paper contains 11 sections, 8 theorems, 40 equations, 2 figures.

Table of Contents

  1. Algebraic Persistence
  2. Persistence Modules
  3. Homomorphisms and Interleavings
  4. Homomorphisms and Interleavings
  5. Kernel, cokernel and the Induced Matching Theorem
  6. The right setup
  7. Persistent cost
  8. Vietoris-Rips filtration and maps
  9. Bounds on the Persistent Cost
  10. Stability
  11. A metric example

Key Result

Theorem 1.5

Let $V_{\bullet},W_{\bullet}$ be p.f.d. persistence modules and $F_{\bullet}: V_{\bullet}\to W_{\bullet}$ a homomorphism. Then there exists an induced matching between intervals in their respective interval decompositions satisfying the following:

Figures (2)

  • Figure 1: A persistence diagram and barcode representation of the same decomposition.
  • Figure 2: Two sets with identical barcodes throughout persistent d-homology which do not admit a $1$-Lipschitz map with small persistent cost.

Theorems & Definitions (22)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5: Induced Matching Theorem, bauer2015inducedbauer2020persistence
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • ...and 12 more