On k-nearest-neighbor regression in metric spaces
László Györfi, Vladimir G. Pestov, Harro Walk, Roi Weiss
Abstract
We study mean squared consistency of k-nearest-neighbor regression for metric space valued features. For a fixed feature distribution mu with atomless distance distributions, we describe how neighbor selection redistributes probability through the density of a uniformly chosen neighbor of an independent query. This gives separate conditions for the approximation error and the conditional response variance, and weakens Stone's domination condition when stronger response moments are available. In particular, when k_n tends to infinity and k_n/n tends to zero, a uniform L2(mu) bound on the selection densities suffices under a finite fourth response moment. We give a corresponding condition in terms of neighborhoods of fixed population mass and recover the Euclidean density bound. Finite global Nagata dimension gives Stone's domination. While sigma-finite Nagata dimensionality is known to ensure universal consistency of k-NN classification in complete separable spaces, we show that this is not the case for mean squared consistency with unbounded responses: we construct a space in this class on which one fixed noiseless response with all finite moments has diverging mean squared error.