Convolution of Persistence Modules

TL;DR

This paper introduces convolution operations for multi-parameter persistence modules, linking them to sheaf theory, and establishes stability and distance measures extending classical interleaving concepts.

Contribution

It defines new convolution operations for persistence modules, relates them to sheaf theory, and develops a convolution distance extending interleaving distance with stability results.

Findings

Convolution operations are isomorphic to derived tensor products.

Formulas for convolutions of interval modules are provided.

Convolution distance extends classical interleaving distance and satisfies stability.

Abstract

We conduct a study of real-valued multi-parameter persistence modules as sheaves and cosheaves. Using the recent work on the homological algebra for persistence modules, we define two different convolution operations between derived complexes of persistence modules. We show that one of these operations is canonically isomorphic to the derived tensor product of graded modules. We give formulas for computing convolutions between single-parameter interval decomposable modules. Our convolution operations are analogous to the convolution of derived complexes of constructible sheaves on $\mathbb{R}^n$ introduced by Schapira and Kashiwara. In our setting, $\mathbb{R}^n$ has a non-standard topology. We show our convolution operation satisfies analogous properties to the convolution of constructible sheaves on $\mathbb{R}^n$ with the standard topology. We define a convolution distance for derived complexes of persistence modules and show that it extends the classical interleaving distance. We also prove stability results from the sheaf and cosheaf points of view.