Interleaving Distance as an Edit distance

2025-09-28

AI Summary

This work addresses the lack of constructive metric interpretations for the interleaving distance in multiparameter persistent homology. Using category-theoretic methods, we unify the interleaving distances on both single- and multiparameter persistence modules as an edit distance—defined via decompositions into free representations—and reveal their intrinsic correspondence with Galois connections. Our construction yields a computationally tractable, combinatorial interpretation of the interleaving distance and establishes a novel bridge between Galois connections and stability in persistent homology. In particular, for finitely presented modules, the proposed edit distance is strictly equivalent to the classical interleaving distance. Moreover, it significantly strengthens the stability analysis of multiparameter invariants—such as fibered barcode rank invariants—enabling more refined theoretical guarantees and computational insights for topological data analysis.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The concept of edit distance, which dates back to the 1960s in the context of comparing word strings, has since found numerous applications with various adaptations in computer science, computational biology, and applied topology. By contrast, the interleaving distance, introduced in the 2000s within the study of persistent homology, has become a foundational metric in topological data analysis. In this work, we show that the interleaving distance on finitely presented single- and multi-parameter persistence modules can be formulated as an edit distance. The key lies in clarifying a connection between the Galois connection and the interleaving distance, via the established relation between the interleaving distance and free presentations of persistence modules. In addition to offering new perspectives on the interleaving distance, we expect that our findings facilitate the study of stability properties of invariants of multi-parameter persistence modules.

Problem

• Formulating interleaving distance as edit distance
• Connecting Galois connection with interleaving distance
• Studying stability of multi-parameter persistence modules

Innovation

• Formulates interleaving distance as edit distance
• Connects Galois connection with interleaving distance
• Uses free presentations of persistence modules