Distances for Multiparameter Persistence Modules

Batan, Mehmet Ali
Persistent homology is an algebraic method to capture the essential topological fea tures of an object. These objects are sometimes a data set called a point cloud or a topological space. After applying filtrations to the data set or topological space, we get persistence modules. One generally computes the interleaving distance between persistence modules to understand the algebraic similarities of these persistence mod ules. 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 fact is known as the isometry theorem. There is no isometry theorem for multiparameter persistence modules, even for special ones. Furthermore, unlike the one-parameter case, interleaving and bottleneck distance computation is not easy, even for special persistence modules. Therefore, we de fine a new distance called steady matching distance and show it is an extended metric for finitely presented interval decomposable persistence modules. Moreover, we investigate the relations between other distances. For interval persis tence modules, we show that the matching distance is equal to the steady matching distance, and the interleaving distance is equal to the bottleneck distance. Moreover, by using the geometry of the underlying intervals of interval persistence modules, we can compute the interleaving distance, which is an upper bound for the steady matching distance. Additionally, we show that the steady matching distance is an upper bound for the matching distance and a lower bound for the bottleneck distance for interval decomposable persistence modules.
Citation Formats
M. A. Batan, “Distances for Multiparameter Persistence Modules,” Ph.D. - Doctoral Program, Middle East Technical University, 2024.