The density of expected persistence diagrams and its kernel based estimation

Abstract

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 [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.

arXiv commentExtended version of a paper published in the proceedings of the Symposium of Computational Geometry 2018

References in corpus (5)

• Statistical topological data analysis using persistence landscapes
• Persistence weighted Gaussian kernel for topological data analysis
• Estimator selection: a new method with applications to kernel density estimation
• Statistical Analysis of Persistence Intensity Functions
• Understanding the Topology and the Geometry of the Space of Persistence Diagrams via Optimal Partial Transport

Cited by in corpus (5)

• A fractal dimension for measures via persistent homology
• Bootstrapping Persistent Betti Numbers and Other Stabilizing Statistics
• Optimal quantization of the mean measure and applications to statistical learning
• Sinkhorn Divergence of Topological Signature Estimates for Time Series Classification
• On the estimation of persistence intensity functions and linear representations of persistence diagrams