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Universal consistency of the $k$-NN rule in metric spaces and Nagata dimension. II

Sushma Kumari, Vladimir G. Pestov

TL;DR

This work probes when the $k$-NN classifier is universally and strongly universally consistent across complete separable metric spaces. It connects learning-theoretic guarantees to geometric/differential properties of the space, notably the Lebesgue–Besicovitch differentiation property and Nagata/de Groot dimensions, extending Stone-type Euclidean results to broader settings. Key contributions include universal and strong universal consistency results in sigma-finite Nagata spaces, non-Archimedean and Nagata-dimension-zero spaces under tailored tie-breaking, and a unified conjecture linking consistency to dimensional and differentiation properties. The findings illuminate how metric geometry governs the reliability of $k$-NN in diverse spaces, with the Heisenberg group serving as a pivotal example separating Nagata and de Groot notions. The revised conjecture sets a roadmap for a comprehensive criterion characterizing universal consistency in arbitrary complete separable metric spaces.

Abstract

We continue to investigate the $k$ nearest neighbour ($k$-NN) learning rule in complete separable metric spaces. Thanks to the results of Cérou and Guyader (2006) and Preiss (1983), this rule is known to be universally consistent in every such metric space that is sigma-finite dimensional in the sense of Nagata. Here we show that the rule is strongly universally consistent in such spaces in the absence of ties. Under the tie-breaking strategy applied by Devroye, Györfi, Krzyżak, and Lugosi (1994) in the Euclidean setting, we manage to show the strong universal consistency in non-Archimedian metric spaces (that is, those of Nagata dimension zero). Combining the theorem of Cérou and Guyader with results of Assouad and Quentin de Gromard (2006), one deduces that the $k$-NN rule is universally consistent in metric spaces having finite dimension in the sense of de Groot. In particular, the $k$-NN rule is universally consistent in the Heisenberg group which is not sigma-finite dimensional in the sense of Nagata as follows from an example independently constructed by Korányi and Reimann (1995) and Sawyer and Wheeden (1992).

Universal consistency of the $k$-NN rule in metric spaces and Nagata dimension. II

TL;DR

This work probes when the -NN classifier is universally and strongly universally consistent across complete separable metric spaces. It connects learning-theoretic guarantees to geometric/differential properties of the space, notably the Lebesgue–Besicovitch differentiation property and Nagata/de Groot dimensions, extending Stone-type Euclidean results to broader settings. Key contributions include universal and strong universal consistency results in sigma-finite Nagata spaces, non-Archimedean and Nagata-dimension-zero spaces under tailored tie-breaking, and a unified conjecture linking consistency to dimensional and differentiation properties. The findings illuminate how metric geometry governs the reliability of -NN in diverse spaces, with the Heisenberg group serving as a pivotal example separating Nagata and de Groot notions. The revised conjecture sets a roadmap for a comprehensive criterion characterizing universal consistency in arbitrary complete separable metric spaces.

Abstract

We continue to investigate the nearest neighbour (-NN) learning rule in complete separable metric spaces. Thanks to the results of Cérou and Guyader (2006) and Preiss (1983), this rule is known to be universally consistent in every such metric space that is sigma-finite dimensional in the sense of Nagata. Here we show that the rule is strongly universally consistent in such spaces in the absence of ties. Under the tie-breaking strategy applied by Devroye, Györfi, Krzyżak, and Lugosi (1994) in the Euclidean setting, we manage to show the strong universal consistency in non-Archimedian metric spaces (that is, those of Nagata dimension zero). Combining the theorem of Cérou and Guyader with results of Assouad and Quentin de Gromard (2006), one deduces that the -NN rule is universally consistent in metric spaces having finite dimension in the sense of de Groot. In particular, the -NN rule is universally consistent in the Heisenberg group which is not sigma-finite dimensional in the sense of Nagata as follows from an example independently constructed by Korányi and Reimann (1995) and Sawyer and Wheeden (1992).
Paper Structure (8 sections, 117 equations, 1 figure)

This paper contains 8 sections, 117 equations, 1 figure.

Table of Contents

  1. Preliminaries: learning rules
  2. Learning in a measurable space
  3. The $k$ nearest neighbour classification rule
  4. Strong consistency
  5. Dimension in the sense of de Groot and the Heisenberg group
  6. Strong consistency in the absence of distance ties
  7. Strong consistency in the non-archimedean case
  8. The revised conjecture

Figures (1)

  • Figure 1: The set $B(x,z,r,b)=B(x,r)\times {\mathbb I} \cup S(x,r)\times N(z,b)$.

Theorems & Definitions (10)

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