Bounding the interleaving distance on concrete categories using a loss function

2026-01-13

AI Summary

Interleaving distance is ubiquitous in computational topological data analysis, yet its NP-hard nature severely limits practical applicability. This work proposes a novel loss-function-based approach that quantifies the deviation between an approximate interleaving morphism and a true interleaving, yielding effective upper bounds on the interleaving distance for generalized persistence modules within a concrete categorical framework. By reformulating the problem into an optimizable form, the method extends the Mapper-graph approach of Chambers et al. to the broader setting of generalized persistent homology. Leveraging tools from category theory and polynomial-time optimization techniques, the framework enables polynomial-time computation of interleaving distance upper bounds under certain conditions—such as for k-parameter persistence modules under specific assumptions—thereby offering a tractable pathway for practical applications.

Technology Category

• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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: Robustness and generalizability of Web mining methods

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.

Problem

• interleaving distance
• topological data analysis
• NP-hard
• generalized persistence modules
• concrete categories

Innovation

• interleaving distance
• loss function
• generalized persistence modules
• concrete categories
• polynomial-time computation