Bayesian Estimation of Topological Features of Persistence Diagrams

Abstract

Persistent homology is a common technique in topological data analysis providing geometrical and topological information about the sample space. All this information, known as topological features, is summarized in persistence diagrams, and the main interest is in identifying the most persisting ones since they correspond to the Betti number values. Given the randomness inherent in the sampling process, and the complex structure of the space where persistence diagrams take values, estimation of Betti numbers is not straightforward. The approach followed in this work makes use of features’ lifetimes and provides a full Bayesian clustering model, based on random partitions, in order to estimate Betti numbers. A simulation study is also presented. An extensive simulation study was done using different synthetic cloud point data in order to understand the performance of the proposed methodology. Based on the scenarios described in Section 4, the sample size will be set to 𝑛 = 300 , 600 , and 900 , and the separation of circles ranges from 1 to 5 for the cases 𝑟 = 2, 3 . For each cloud point data, 𝑠 = 100 replications were simulated, and each estimator for 𝛽 0 was computed.