A nearest neighbor characterization of Lebesgue points in metric measure spaces

  • Tommaso R. Cesari

    Toulouse School of Economics, Toulouse, France
  • Roberto Colomboni

    Università degli Studi di Milano and Istituto Italiano di Tecnologia, Milano, Italy
A nearest neighbor characterization of Lebesgue points in metric measure spaces cover
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Abstract

The property of almost every point being a Lebesgue point has proven to be crucial for the consistency of several classification algorithms based on nearest neighbors.We characterize Lebesgue points in terms of a 1-nearest neighbor regression algorithm for pointwise estimation, fleshing out the role played by tie-breaking rules in the corresponding convergence problem. We then give an application of our results, proving the convergence of the risk of a large class of 1-nearest neighbor classification algorithms in general metric spaces where almost every point is a Lebesgue point.

Cite this article

Tommaso R. Cesari, Roberto Colomboni, A nearest neighbor characterization of Lebesgue points in metric measure spaces. Math. Stat. Learn. 3 (2020), no. 1, pp. 71–112

DOI 10.4171/MSL/19