The density of expected persistence diagrams and its kernel based estimation
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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 R^2 that can equivalently be seen as discrete measures in 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 Čech and Rips-Vietoris filtrations, the expected persistence diagram, that is a deterministic measure on 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 & al., Persistence images: a stable vector representation of persistent homology] 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.
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