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

28 Feb 2020  ·  , , ·

The $k$ nearest neighbour learning rule (under the uniform distance tie breaking) is universally consistent in every metric space $X$ that is sigma-finite dimensional in the sense of Nagata. This was pointed out by C\'erou and Guyader (2006) as a consequence of the main result by those authors, combined with a theorem in real analysis sketched by D. Preiss (1971) (and elaborated in detail by Assouad and Quentin de Gromard (2006)). We show that it is possible to give a direct proof along the same lines as the original theorem of Charles J. Stone (1977) about the universal consistency of the $k$-NN classifier in the finite dimensional Euclidean space. The generalization is non-trivial because of the distance ties being more prevalent in the non-euclidean setting, and on the way we investigate the relevant geometric properties of the metrics and the limitations of the Stone argument, by constructing various examples.

PDF Abstract

## Code Add Remove Mark official

No code implementations yet. Submit your code now

## Datasets

Add Datasets introduced or used in this paper

## Results from the Paper Add Remove

Submit results from this paper to get state-of-the-art GitHub badges and help the community compare results to other papers.