The Density of Expected Persistence Diagrams and its Kernel Based Estimation

Persistence diagrams play a fundamental role in Topological Data Analysis where they are used as topological descriptors of filtrations built on top of data. They consist in discrete multisets of points in the plane R2R2\mathbb{R}^2 that can equivalently be seen as discrete measures in R2R2\mathbb{R}^2. When the data come as a random point cloud, these discrete measures become random measures whose expectation is studied in this paper. First, we show that for a wide class of filtrations, including the \v Cech and Rips-Vietoris filtrations, the expected persistence diagram, that is a deterministic measure on R2R2\mathbb{R}^2, has a density with respect to the Lebesgue measure. Second, building on the previous result we show that the persistence surface recently introduced in [H. Adams et al., Journal of Machine Learning Research 18.8 (2017)] can be seen as a kernel estimator of this density. We propose a cross-validation scheme for selecting an optimal bandwidth, which is proven to be a consistent procedure to estimate the density.