Algebraic topology in machine learning

This dissertation presents novel approaches and applications of machine learning architectures. In particular, these approaches are based on tools from topological data analysis and are used in conjunction with conventional machine learning methods. Topological data analysis, which is based on algebraic topology, can identify significant global mathematical structures which are out of reach of many other approaches. When we use topology we benefit from generality, and when we use conventional methods we benefit from specificity.

This dissertation contains a broad overview of data science and topological data analysis, then transitions to three distinct machine learning applications of these methods. The first application uses linear methods to discover the inherent dimensionality of the manifold given by congressional roll call votes. The second uses persistent homology to identify extremely noisy images in both supervised and unsupervised tasks. The last application uses mapper objects to produce robust classification algorithms. Two additional projects are presented later in the appendix, and are related to the three main applications. The first of these constructs a method to choose optimal optimizers, and the second places mathematical constraints on the structure of renormalization group flows.