Smooth Summaries of Persistence Diagrams and Texture Classification

Topological data analysis (TDA) is a rising field in the intersection of mathematics, statistics, and computer science/data science. Persistent homology is one of the most commonly used tools in TDA, in part because it can be easily visualized in the form of a persistence diagram. However, performing machine learning algorithms directly on persistence diagrams is a challenging task, and so a number of summaries have been proposed which transform persistence diagrams into vectors or functions. Many of these summaries fall into the persistence curve framework developed by Chung and Lawson. We extend this framework and introduce new class of smooth persistence curves which we call Gaussian persistence curves. We investigate the statistical properties of Gaussian persistence curves and apply them to texture datasets: UIUCTex and KTH. Our classification results on these texture datasets perform competitively with the current state-of-arts methods in TDA.