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 $\mathbb{R}^2$ that can equivalently be seen as discrete measures in $\mathbb{R}^2$. When the data is assumed to be random, 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 Cech and Vietoris-Rips filtrations, but also the sublevels of a Brownian motion, the expected persistence diagram, that is a deterministic measure on $\mathbb{R}^2$ , has a density with respect to the Lebesgue measure. Second, building on the previous result we show that the persistence surface recently introduced in Adams et al. [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.