The density of expected persistence diagrams and its kernel based estimation. (English) Zbl 1473.55002
Speckmann, Bettina (ed.) et al., 34th international symposium on computational geometry, SoCG 2018, June 11–14, 2018, Budapest, Hungary. Wadern: Schloss Dagstuhl – Leibniz Zentrum für Informatik. LIPIcs – Leibniz Int. Proc. Inform. 99, Article 26, 15 p. (2018).
Summary: 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 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 \(\mathbb{R}^2\), 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., J. Mach. Learn. Res. 18, Paper No. 8, 35 p. (2017; Zbl 1431.68105)] 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.
For the entire collection see [Zbl 1390.68027].
For the entire collection see [Zbl 1390.68027].
MSC:
| 55N31 | Persistent homology and applications, topological data analysis |
| 68U05 | Computer graphics; computational geometry (digital and algorithmic aspects) |
Citations:
Zbl 1431.68105Software:
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