A Nearest Neighbor Characterization of Lebesgue Points in Metric Measure Spaces
Machine Learning
2021-01-13 v4 Probability
Machine Learning
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.
Keywords
Cite
@article{arxiv.2007.03937,
title = {A Nearest Neighbor Characterization of Lebesgue Points in Metric Measure Spaces},
author = {Tommaso Cesari and Roberto Colomboni},
journal= {arXiv preprint arXiv:2007.03937},
year = {2021}
}