Persistence Diagram Cardinality and its Applications

This dissertation studies persistence diagrams and their usefulness in machine learning. Persistence diagrams are summaries of underlying topological structure present within data. These diagrams are especially applicable for analyzing data whose shape is a relevant descriptor, as they provide unique information as sets instead of vectors. Although there are methods for vectorizing persistence diagrams, we focus instead on statistical learning schemes that use persistence diagrams directly. In particular, the cardinality of the diagrams proves itself a useful indicator, although this cardinality is variable at higher dimensions. To better understand and use the cardinality of persistence diagrams, we prove that the cardinality is bounded using both statistics and geometry. We also prove stability of a cardinality-based persistence diagram distance in a continuous fashion. These results are then applied to analyze persistence diagrams generated from structures of materials.We also develop a Bayesian framework for modeling the cardinality and spatial distributions of points within persistence diagrams using i.i.d. cluster point processes. From Gaussian mixture and binomial priors, we derive equations for the posterior cardinality and spatial distributions. We also introduce a distribution to account for noise typical of persistence diagrams. This framework provides a means of classifying biochemical networks present within cells using Bayes factors. We also provide a favorable comparison of the Bayesian classification of networks with several other methods.