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$\ell^p$-Stability of Weighted Persistence Diagrams

Aziz Burak Gülen, Facundo Mémoli, Amit Patel

TL;DR

This work develops a theory of weighted persistence diagrams by embedding finite metric measure spaces (mm-spaces) into a functorial pipeline that yields weighted filtrations and weighted persistence diagrams. It proves $p$-stability of these weighted diagrams under the $p$-edit distance, controlled by the $p$-Gromov–Wasserstein distance, leveraging a robust functorial framework and an OT-like interpretation. The approach extends existing unweighted persistence results by introducing a weighted, measure-aware perspective that captures global distance distributions and enhances discriminative power. The results provide both theoretical stability guarantees and practical insight into when weights improve topological summaries for comparing complex data. Overall, the paper unifies category-theoretic, geometric, and transport viewpoints to advance the stability and applicability of weighted persistence in TDA.

Abstract

We introduce the concept of weighted persistence diagrams and develop a functorial pipeline for constructing them from finite metric measure spaces. This builds upon an existing functorial framework for generating classical persistence diagrams from finite pseudo-metric spaces. To quantify differences between weighted persistence diagrams, we define the $p$-edit distance for $p\in [1,\infty]$, and-focusing on the weighted Vietoris-Rips filtration-we establish that these diagrams are stable with respect to the $p$-Gromov-Wasserstein distance as a direct consequence of functoriality. In addition, we present an Optimal Transport-inspired formulation of the $p$-edit distance, enhancing its conceptual clarity. Finally, we explore the discriminative power of weighted persistence diagrams, demonstrating advantages over their unweighted counterparts.

$\ell^p$-Stability of Weighted Persistence Diagrams

TL;DR

This work develops a theory of weighted persistence diagrams by embedding finite metric measure spaces (mm-spaces) into a functorial pipeline that yields weighted filtrations and weighted persistence diagrams. It proves -stability of these weighted diagrams under the -edit distance, controlled by the -Gromov–Wasserstein distance, leveraging a robust functorial framework and an OT-like interpretation. The approach extends existing unweighted persistence results by introducing a weighted, measure-aware perspective that captures global distance distributions and enhances discriminative power. The results provide both theoretical stability guarantees and practical insight into when weights improve topological summaries for comparing complex data. Overall, the paper unifies category-theoretic, geometric, and transport viewpoints to advance the stability and applicability of weighted persistence in TDA.

Abstract

We introduce the concept of weighted persistence diagrams and develop a functorial pipeline for constructing them from finite metric measure spaces. This builds upon an existing functorial framework for generating classical persistence diagrams from finite pseudo-metric spaces. To quantify differences between weighted persistence diagrams, we define the -edit distance for , and-focusing on the weighted Vietoris-Rips filtration-we establish that these diagrams are stable with respect to the -Gromov-Wasserstein distance as a direct consequence of functoriality. In addition, we present an Optimal Transport-inspired formulation of the -edit distance, enhancing its conceptual clarity. Finally, we explore the discriminative power of weighted persistence diagrams, demonstrating advantages over their unweighted counterparts.

Paper Structure

This paper contains 36 sections, 24 theorems, 164 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Previous Work
  3. Our Contributions
  4. Acknowledgements
  5. Preliminaries
  6. Posets and Galois Connections
  7. Möbius Inversion.
  8. Simplicial Complexes and Filtrations
  9. Edit Distance
  10. Metric Spaces
  11. Gromov-Hausdorff Distance.
  12. Measure Spaces
  13. Metric Measure Spaces
  14. Gromov-Wasserstein Distance.
  15. and as Edit Distances
  16. ...and 21 more sections

Key Result

Theorem 1.1.1

Let $\mathsf{F}$ and $\mathsf{G}$ be two filtrations in $\mathsf{Fil}(K)$ and let $\mathsf{PD}_d^\mathsf{F}$ and $\mathsf{PD}_d^\mathsf{G}$ denote their degree-$d$ persistence diagrams respectively. Then,

Figures (8)

  • Figure 1: Two ultrametric spaces $X$ and $Y$, each with 4 points, shown in the figure (through their dendrogram representations) have the same Vietoris-Rips persistence diagrams in all degrees. However, when regarded as mm-spaces (by endowing them with their respective uniform measures), their weighted Vietoris-Rips persistence diagrams are different; see \ref{['ex:ums']}.
  • Figure 2: The histograms of distances in $(X,d_X, \mu_X)$ (on the left) and in $(Y,d_Y, \mu_Y)$ (on the right) described in \ref{['ex:ums']}.
  • Figure 3: Two mm-spaces with the same degree-1 persistence diagrams but with different weighted persistence diagrams.
  • Figure 4: Degree-1 persistence diagrams of the Vietoris-Rips filtrations of $(X,d_X)$ and $(Y,d_Y)$ described in \ref{['ex: same pd-1 different wpd-1']}.
  • Figure 5: Two mm-spaces with the same (flipped) global distributions of distances but with different weighted persistence diagrams.
  • ...and 3 more figures

Theorems & Definitions (96)

  • Theorem 1.1.1: mccleary2022edit
  • Theorem 1.2.1
  • Theorem 1.2.2
  • Definition 2.1.1: Pushforward and pullback
  • Theorem 2.1.2: Rota's Galois Connection Theorem (RGCT) gal-conn
  • Definition 2.3.1: Edit distance
  • Proposition 2.3.2: Functorial edit distance stability
  • proof
  • Definition 2.4.1: Distortion
  • Definition 2.4.2: Product
  • ...and 86 more