Topics in Geometric and Topological Data Analysis

Geometric and topological data analysis have recently attained much interests from the statistics community for their capabilities to perform inferential tasks on types of data that traditional statistical disciplines had not been able to handle and for their utility that enables to capture certain aspects of the data that had not been observed. My dissertation develops several methods that focus on various aspects across the two disciplines. In the geometric realm, I developed efficient numerical routines for maximum likelihood estimation of the spherical normal distribution that was recently proposed to replace standard Euclidean discrepancy in the celebrated von Mises-Fisher distribution with the geodesic distance that better reflects intrinsic geometry of the unit hypersphere. On the topological domain, I first proposed a systematic approach to compare multiple latent space embeddings for networks of potentially varying size and invariant under label permutation in light of deriving inferential insights on how shapes of networks differ via clustering and hypothesis testing procedures. Lastly, I developed a topology-based linear dimensionality reduction algorithm that minimizes the discrepancy of topological properties between the data and its projection. I also proposed theoretical concepts for two measures that quantify topological equivalence of the data and its embedding induced by any linear maps.