A Nearest Neighbor Characterization of Lebesgue Points in Metric Measure Spaces

TL;DR

This paper characterizes Lebesgue points in metric measure spaces using 1-Nearest Neighbor regression and demonstrates the convergence of certain nearest neighbor classifiers' risk in these spaces.

Contribution

It provides a new characterization of Lebesgue points via 1-NN regression and analyzes tie-breaking rules' impact on convergence.

Findings

Characterization of Lebesgue points through 1-NN regression

Convergence proof of NN classification risk in metric spaces

Role of tie-breaking rules in convergence analysis

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.