A roadmap for the computation of persistent homology

• Published in: EPJ data science Vol. 6; no. 1; pp. 17 - 38
• Main Authors: Otter, Nina, Porter, Mason A, Tillmann, Ulrike, Grindrod, Peter, Harrington, Heather A
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 09.08.2017; Springer Nature B.V; SpringerOpen
• Subjects: Algorithms; Benchmarks; Complexity; Computation; Computer Appl. in Social and Behavioral Sciences; Computer applications; Computer Science; Data analysis; Data-driven Science; Datasets; Homology; Modeling and Theory Building; networks; persistent homology; point-cloud data; Qualitative analysis; Regular; Regular Article; topological data analysis; point-cloud data; networks; persistent homology; topological data analysis
• ISSN: 2193-1127; 2193-1127
• DOI: 10.1140/epjds/s13688-017-0109-5

Persistent homology (PH) is a method used in topological data analysis (TDA) to study qualitative features of data that persist across multiple scales. It is robust to perturbations of input data, independent of dimensions and coordinates, and provides a compact representation of the qualitative features of the input. The computation of PH is an open area with numerous important and fascinating challenges. The field of PH computation is evolving rapidly, and new algorithms and software implementations are being updated and released at a rapid pace. The purposes of our article are to (1) introduce theory and computational methods for PH to a broad range of computational scientists and (2) provide benchmarks of state-of-the-art implementations for the computation of PH. We give a friendly introduction to PH, navigate the pipeline for the computation of PH with an eye towards applications, and use a range of synthetic and real-world data sets to evaluate currently available open-source implementations for the computation of PH. Based on our benchmarking, we indicate which algorithms and implementations are best suited to different types of data sets. In an accompanying tutorial, we provide guidelines for the computation of PH. We make publicly available all scripts that we wrote for the tutorial, and we make available the processed version of the data sets used in the benchmarking.