A Closed Formula for the Interleaving Distance of Rectangle Persistence Modules

Tuesday, 8-Oct-2024 

Abstract

Persistent homology is an algebraic method to capture the essential topological features of an object. These objects are sometimes a data set called a point cloud or a topological space. After applying filtration to the data set or topological space, we get an essential object in persistence theory, generally called the persistence module. One typically computes the interleaving distance between persistence modules to understand the algebraic similarities of these persistence modules. In addition to the interleaving distance, the bottleneck distance can be computed between the barcodes of these persistence modules. For one-parameter persistence modules, the interleaving distance equals the bottleneck distance. This important fact is known as the isometry theorem. There is no isometry theorem for multiparameter persistence modules, even for special persistence modules such as rectangle decomposable.

Furthermore, unlike the one-parameter case, interleaving and bottleneck distance computation is not easy, even for special persistence modules. Therefore, we give a closed formula for calculating the interleaving distance between rectangle persistence modules that depend solely on the geometry of the underlying rectangles. Moreover, we extend our results to calculate the bottleneck distance for rectangle decomposable persistence modules.