Statistical Learning and Analysis on Homology-Based Features

Abstract: Stable rank has recently been proposed as an invariant to encode the result of persistent homology, a method used in topological data analysis. In this thesis we develop methods for statistical analysis as well as machine learning methods based on stable rank. As stable rank may be viewed as a mapping to a Hilbert space, a kernel can be constructed from the inner product in this space. First, we investigate this kernel in the context of kernel learning methods such as support-vector machines. Next, using the theory of kernel embedding of probability distributions, we give a statistical treatment of the kernel by showing some of its properties and develop a two-sample hypothesis test based on the kernel. As an alternative approach, a mapping to a Euclidean space with learnable parameters can be conceived, serving as an input layer to a neural network. The developed methods are first evaluated on synthetic data. Then the two-sample hypothesis test is applied on the OASIS open access brain imaging dataset. Finally a graph classification task is performed on a dataset collected from Reddit.