Approximating Continuous Functions on Persistence Diagrams Using Template Functions
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The persistence diagram is an increasingly useful tool arising from the field of Topological Data Analysis. However, using these diagrams in conjunction with machine learning techniques requires some mathematical finesse. The most success to date has come from finding methods for turning persistence diagrams into vectors in R^n in a way which preserves as much of the space of persistence diagrams as possible, commonly referred to as featurization. In this paper, we describe a mathematical framework for featurizing the persistence diagram space using template functions. These functions are general as they are only required to be continuous, have a compact support, and separate points. We discuss two example realizations of these functions: tent functions and Chybeyshev interpolating polynomials. Both of these functions are defined on a grid superposed on the birth-lifetime plane. We then combine the resulting features with machine learning algorithms to perform supervised classification and regression on several example data sets, including manifold data, shape data, and an embedded time series from a Rossler system. Our results show that the template function approach yields high accuracy rates that match and often exceed the results of existing methods for featurizing persistence diagrams. One counter-intuitive observation is that in most cases using interpolating polynomials, where each point contributes globally to the feature vector, yields significantly better results than using tent functions, where the contribution of each point is localized to its grid cell. Along the way, we also provide a complete characterization of compact sets in persistence diagram space endowed with the bottleneck distance.
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