An isometry theorem induced by the Radon transform between the convolution and interleaving distances

TL;DR

This paper establishes an isometry theorem linking convolution and interleaving distances for sheaves via the Radon transform, advancing the understanding of multi-parameter persistence modules in data analysis.

Contribution

It introduces an isometry theorem induced by the Radon transform that relates convolution and interleaving distances for sheaves, providing a new theoretical tool for multi-parameter persistence modules.

Findings

Radon transform induces an isometry between convolution and interleaving distances.

Developed convolution and interleaving distances on localized categories of sheaves.

Radon transform simplifies multi-directional movements to one-directional, facilitating analysis.

Abstract

One-parameter persistence modules are applied to various subjects as tools in data analysis. On the other hand, since the theoretical study of multi-parameter persistence modules is not enough and in progress, they have few applications. The sheaf theory is expected to elucidate detailed properties of persistence modules and give features of multi-parameter ones for applications. However, the categories of sheaves on two or more dimensional Euclidean spaces have more complicated structures than those on R. The Radon transform for sheaves is a useful dimension reduction technique and induces a categorical equivalence between the localized bounded derived categories of sheaves. We show We develop the convolution and the interleaving distances on these localized categories by improving original distances on the derived categories of sheaves. The convolution bifunctor defines these distances. We show that the Radon transform changes multi-directional movements given by the convolution bifunctor to one-directional movements and induce an isometry theorem between these distances.