Minimal Cycle Representatives in Persistent Homology Using Linear Programming: An Empirical Study With User’s Guide

Journal article

Persistent homology (PH) is a method for generating topology-inspired representations of data. Empirical studies that investigate the properties of PH, such as its sensitivity to perturbations or ability to detect a feature of interest, commonly rely on training and testing an additional model on the basis of the PH representation. To gain more intrinsic insights about PH, independently of the choice of such a model, we propose a novel methodology based on the pull-back geometry that a PH encoding induces on the data manifold. The spectrum and eigenvectors of the induced metric help to identify the most and least significant information captured by PH. Furthermore, the pull-back norm of tangent vectors provides insights about the sensitivity of PH to a given perturbation, or its potential to detect a given feature of interest, and in turn its ability to solve a given classification or regression problem. Experimentally, the insights gained through our methodology align well with the existing knowledge about PH. Moreover, we show that the pull-back norm correlates with the performance on downstream tasks, and can therefore guide the choice of a suitable PH encoding

Authors: Li, L; Thompson, C; Henselman-Petrusek, G; Giusti, C; Ziegelmeier, L

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Frontiers Media
• Journal: Frontiers in Artificial Intelligence
• Volume: 4
• Pages: 681117-681117
• Publication date: 2021-10-11
• DOI: 10.3389/frai.2021.681117
• EISSN: 2624-8212
• ISSN: 2624-8212
• Language: English
• Keywords: Solver; Algorithm; Context (archaeology); Mathematical optimization; Computation; Integer programming; Integer (computer science); Computer science; Mathematics; Linear programming