Interleaving Distance as a Galois-Edit Distance
This work provides a new theoretical perspective on a foundational metric in topological data analysis, but the result is primarily conceptual and does not introduce new computational methods or empirical improvements.
The paper shows that the interleaving distance on finitely presented single- and multi-parameter persistence modules can be formulated as a Galois-edit distance, and uses this formulation to provide an alternative proof of the bottleneck stability theorem.
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 a so-called Galois-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 will facilitate the study of stability properties of invariants for multi-parameter persistence modules. As an application of the Galois-edit formulation of the interleaving distance, we present an alternative proof of the well-known bottleneck stability theorem.