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Edit Distance and Persistence Diagrams Over Lattices

Alexander McCleary, Amit Patel

TL;DR

The paper generalizes persistent diagrams to filtrations indexed by finite metric lattices and presents a functorial, stable pipeline built from three categories: ${\mathsf{Fil}}(K)$, ${\mathsf{Mon}}$, and ${\mathsf{Fnc}}$. It introduces the birth–death functor ${\mathsf{ZB}}_*$ and the Möbius inversion functor ${\mathsf{MI}}$, proving both are $1$-Lipschitz and that the resulting diagrams are stable invariants. The authors show that the generalized bottleneck distance is strongly equivalent to the edit distance within this framework, linking classical and generalized persistence. This work enables robust applications in homological inference and machine learning by providing a principled, functorial, and stable way to extract and compare multi-parameter persistence information via Möbius inversion on lattices.

Abstract

We build a functorial pipeline for persistent homology. The input to this pipeline is a filtered simplicial complex indexed by any finite metric lattice and the output is a persistence diagram defined as the Möbius inversion of its birth-death function. We adapt the Reeb graph edit distance to each of our categories and prove that both functors in our pipeline are $1$-Lipschitz making our pipeline stable. Our constructions generalize the classical persistence diagram and, in this setting, the bottleneck distance is strongly equivalent to the edit distance.

Edit Distance and Persistence Diagrams Over Lattices

TL;DR

The paper generalizes persistent diagrams to filtrations indexed by finite metric lattices and presents a functorial, stable pipeline built from three categories: , , and . It introduces the birth–death functor and the Möbius inversion functor , proving both are -Lipschitz and that the resulting diagrams are stable invariants. The authors show that the generalized bottleneck distance is strongly equivalent to the edit distance within this framework, linking classical and generalized persistence. This work enables robust applications in homological inference and machine learning by providing a principled, functorial, and stable way to extract and compare multi-parameter persistence information via Möbius inversion on lattices.

Abstract

We build a functorial pipeline for persistent homology. The input to this pipeline is a filtered simplicial complex indexed by any finite metric lattice and the output is a persistence diagram defined as the Möbius inversion of its birth-death function. We adapt the Reeb graph edit distance to each of our categories and prove that both functors in our pipeline are -Lipschitz making our pipeline stable. Our constructions generalize the classical persistence diagram and, in this setting, the bottleneck distance is strongly equivalent to the edit distance.

Paper Structure

This paper contains 22 sections, 24 theorems, 60 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Outline
  4. Applications of Stability
  5. Homological Inference
  6. Machine Learning
  7. Preliminaries
  8. Lattices
  9. Metric Lattices
  10. Filtrations
  11. Monotone Integral Functions
  12. Birth-Death Functor
  13. Integral Functions
  14. Möbius Inversion Functor
  15. Edit Distance
  16. ...and 7 more sections

Key Result

Proposition 3.1

Let $P$ and $Q$ be finite lattices and $\alpha : P \to Q$ a bounded lattice function. Then for all $a \in Q$, the pre-image $\alpha^{-1} [\bot, a]$ has a maximal element.

Figures (10)

  • Figure 1: Hasse diagrams of two finite metric lattices $P$ and $Q$. The metrics $d_P$ and $d_Q$ assigns to every pair of elements the length (i.e. the number of edges) of the shortest path between them. For example, $d_P(a,d) = 2$ and $d_Q(p, q) = 1$. The function $\alpha : P \to Q$ defined as $\alpha(a) = \alpha(b) = p$ and $\alpha(c) = \alpha(d) = r$ is a bounded lattice function. The distortion of $\alpha$ is $|| \alpha || = 1$.
  • Figure 2: Hasse diagrams of the lattices $\bar{P}$ and $\bar{Q}$ where $P$ and $Q$ are from Example \ref{['ex:one']}. The morphism $\alpha : P \to Q$ from the same example extends to the morphism $\bar{\alpha} : \bar{P} \to \bar{Q}$ as follows. The function $\bar{\alpha}$ sends $\{ [a,a], [a,b], [b,b] \}$ to $\{ [p,p] \}$, $\{ [a,c], [a,d], [b,d] \}$ to $\{ [p,r] \}$, and $\{ [c,c], [c,d], [d,d] \}$ to $\{ [r,r] \}$. The distortion of $\bar{\alpha}$ is $|| \bar{\alpha} || = || \alpha || = 1$.
  • Figure 3: Filtrations $F$ and $G$ of the $2$-simplex along with a filtration-preserving morphism $\alpha$ as described in Example \ref{['ex:one']}.
  • Figure 4: Two monotone integral functions $f$ and $g$ on the metric lattices $\bar{P}$ and $\bar{Q}$ from Example \ref{['ex:three']}. The triple $(f, g, \bar{\alpha})$, where $\bar{\alpha} : \bar{P} \to \bar{Q}$ is from the same example, is a monotone-preserving morphism from $f$ to $g$.
  • Figure 5: Two integral functions $\sigma$ and $\tau$ on $\bar{P}$ and $\bar{Q}$, respectively, from Example \ref{['ex:three']}. The bounded lattice function $\bar{\alpha} : \bar{P} \to \bar{Q}$ from the same figure is a charge-preserving morphism from $\sigma$ to $\tau$.
  • ...and 5 more figures

Theorems & Definitions (81)

  • Proposition 3.1
  • proof
  • Example 3.2
  • Lemma 3.3
  • Proposition 3.4
  • proof
  • Example 3.5
  • Definition 4.1
  • Definition 4.2
  • Remark 4.3
  • ...and 71 more