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Abstract:
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We study the problems of reconstructing a set S and of estimating one its topological characteristic, namely its number of connected components, from random points of S drawn from some probability measure. We focus on the certainly most simple set estimator defined as the union of balls centered at the random points. Additionally, we propose a graph-based estimator of the number of connected components of S. Using tools from Riemannian geometry, and under mild analytic conditions on the underlying density of the data, we derive the exact rate of convergence of this set estimator and prove the consistency of the estimator of the number of connected components. Statistical applications include density support estimation and estimation of the number of clusters in data partitioning.
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