Bounding the interleaving distance on concrete categories using a loss function

TL;DR

This paper introduces a method to bound the interleaving distance in topological data analysis using a loss function, enabling polynomial-time computation in certain cases despite the NP-hardness of the problem.

Contribution

It generalizes the bounding of interleaving distance to concrete categories using a loss function, extending prior work on mapper graphs.

Findings

Loss function provides a measurable bound on interleaving distance.

Polynomial-time computation possible under specific conditions.

Applicable to various persistence modules and topological structures.

Abstract

The interleaving distance is arguably the most widely used metric in topological data analysis (TDA) due to its applicability to a wide array of inputs of interest, such as (multiparameter) persistence modules, Reeb graphs, merge trees, and zigzag modules. However, computation of the interleaving distance in the vast majority of this settings is known to be NP-hard, limiting its use in practical settings. Inspired by the work of Chambers et al. on the interleaving distance for mapper graphs, we solve a more general problem bounding the interleaving distance between generalized persistence modules on concrete categories via a loss function. This loss function measures how far an assignment, which can be thought of as an interleaving that might not commute, is from defining a true interleaving. We give settings for which the loss can be computed in polynomial time, including for certain assumptions on $k$-parameter persistence modules.