Some Topics in Topological Data Analysis - Di, Yang (2021)
In recent years there has been growing interest within Statistics in topological aspects of random objects, one important direction being Topological Data Analysis (TDA) and the associated concept of Persistent Homology. This research aims to investigate both theoretical and computational aspects of TDA. In the first strand of this research the aim is to generalize the central limit theorem (CLT) given by Kahle and Meckes (2013, 2015) for Betti numbers in Erdős–Rényi random graphs, to a CLT for Betti numbers in the stochastic block model. In addressing this problem, we have provided results on the spectral structure of the adjacency matrix and the normalized graph Laplacian in stochastic block models which appear to be new. The second strand of the research is to investigate numerically the relationship between the topological summaries computed under the full sample and under subsamples. Subsampling often needs to be considered because existing computational algorithms for TDA tend to break down for larger sample sizes as computational demands grow rapidly with sample size. One important finding is that subsampling which exploits existing structure in the data is likely to do much better than purely random subsampling. In this PhD thesis, numerical results are given for various types of simulated data through to real datasets.