Machine Learning with Topological Data Analysis

Topological Data Analysis (TDA) is a relatively new focus in the fields of statistics and machine learning. Methods of exploiting the geometry of data, such as clustering, have proven theoretically and empirically invaluable. TDA provides a general framework within which to study topological invariants (shapes) of data, which are more robust to noise and can recover information on higher dimensional features than immediately apparent in the data. A common tool for conducting TDA is persistence homology, which measures the significance of these invariants. Persistence homology has prominent realizations in methods of data visualization, statistics and machine learning. Extending ML with TDA and persistence homology offers greater interpretability and generalizability than can be achieved with current state of the art tools. My work explores several applications of TDA and persistence homology. TDA has previously been used to track the movements of intracellular bodies and it follows that it could also be used as a tool to examine the cytoskeleton of cells as a network captured in confocal microscopic images. I show recent and developing work in an application of persistence homology for the classification of intra-cellular networks. In addition to persistence homology, methods from TDA can be used to partition data into hierarchical clusters. I explore empirical and theoretical findings from topological clustering of images and graphs in order to propose a novel generalization on the convolutional neural network (CNN) and contrast its empirical results with the traditional CNN.