Geometry of Stratified Spaces for the Analysis of Complex Data

Abstract

Traditionally, statistical data has been in the form of elements of Euclidean space. However, as data complexity increases, it is assumed to lie on lower dimensional non-linear space such as smooth manifolds, in what is known as the manifold hypothesis. Nonetheless, real-world data is not always in the form of smooth manifolds but, in general, can lie on stratified spaces. In this thesis, we explore the geometry of stratified spaces with the overall objective of enabling statistics on these spaces. More specifically, we provide answers to the following two problems.A fundamental task in object recognition is to identify when two shapes are similar. One approach to rendering this as a precise mathematical problem is to look at the space of all shapes and define a metric on it. This approach has been taken by renowned statisticians and mathematicians like Kendall, Grenander, Mumford, Michor, and others. In this thesis, we provide an algebraic construction of the moduli space of shapes and define metrics on it with the objective of developing a statistical theory on shapes. The construction is far more general than existing constructions, as it doesn't restrict `shapes' to smooth manifolds and includes a broad category of spaces, including many stratified spaces. The foundation of this construction relies on the topological analogue of the Radon transform, building on the work of Schapira who showed that such transforms are injective.This thesis also provides a starting point for developing a theory of diffusion processes on general stratified spaces. On Euclidean spaces, Brownian motion is constructed by taking scaled limits of random walks. This approach is challenging because stratified spaces are not only non-linear and lack addition but also the tangent spaces of stratified spaces are non-linear, unlike smooth manifolds. So, instead, we define Brownian motion on stratified spaces by taking appropriate limits of Dirichlet forms. Sturm took this approach for general metric measure spaces, where he came up with a measure-theoretic condition required for these Dirichlet forms to converge properly. We prove this is the case for certain compact subanalytic spaces.Parts of the thesis are based on joint work with Justin Curry and Sayan Mukherjee.