Differential Calculus and Optimization in Persistence Module Categories

TL;DR

This paper introduces a framework for differential calculus and optimization within persistence module categories, leveraging their stability and approximability properties to enable mathematical analysis and computations in topological data analysis.

Contribution

It develops a novel calculus and optimization framework for persistence modules, facilitating advanced analysis and applications in topological data analysis and related fields.

Findings

Framework provides convergence guarantees for calculus operations

Enables optimization techniques in persistence module categories

Enhances mathematical tools for topological data analysis

Abstract

Persistence modules are representations of products of totally ordered sets in the category of vector spaces. They appear naturally in the representation theory of algebras, but in recent years they have also found applications in other areas of mathematics, including symplectic topology, complex analysis, and topological data analysis, where they arise from filtrations of topological spaces by the sublevel sets of real-valued functions. Two fundamental properties of persistence modules make them useful in such contexts: (1) the fact that they are stable under perturbations of the originating functions, and (2) the fact that they can be approximated, in the sense of relative homological algebra, by classes of indecomposable modules with an elementary structure. In this text we give an introduction to the theory of persistence modules, then we explain how the above properties can be leveraged to build a framework for differential calculus and optimization with convergence guarantees in persistence module categories.