Persistence and the Sheaf-Function Correspondence

TL;DR

This paper explores the relationship between distances on constructible functions and sheaf theory, revealing that such distances are trivial unless distinguished by the Euler integral, impacting invariants in Topological Data Analysis.

Contribution

It characterizes which distances on constructible functions are controlled by the convolution distance via the sheaf-function correspondence, showing they are essentially trivial.

Findings

Distances controlled by the convolution distance vanish unless functions differ in Euler integral.

Implication that non-trivial additive invariants of persistence modules cannot be continuous for the interleaving distance.

Establishes a fundamental limitation for invariants in Topological Data Analysis.

Abstract

The sheaf-function correspondence identifies the group of constructible functions on a real analytic manifold $M$ with the Grothendieck group of constructible sheaves on $M$. When $M$ is a finite dimensional real vector space, Kashiwara-Schapira have recently introduced the convolution distance between sheaves of $k$-vector spaces on $M$. In this paper, we characterize distances on the group of constructible functions on a real finite dimensional vector space that can be controlled by the convolution distance through the sheaf-function correspondence. Our main result asserts that such distances are almost trivial: they vanish as soon as two constructible functions have the same Euler integral. We formulate consequences of our result for Topological Data Analysis: there cannot exists non-trivial additive invariants of persistence modules that are continuous for the interleaving distance.