Section: New Results

Rates of convergence of the functional k –nearest neighbor estimate

Participants : Frédéric Cérou, Arnaud Guyader.

See  3.5

This is a collaboration with Gérard Biau, from université Pierre et Marie Curie, ENS Paris and INRIA Paris Rocquencourt (project–team CLASSIC).

Motivated by a broad range of potential applications, such as regression on curves, we investigate rates of convergence of the k -nearest neighbor estimate of the regression function, based on N independent copies of the pair (X, Y) , when X is in a suitable ball in some functional space. Using compact embedding theory, we present explicit and general finite sample bounds on the expected squared difference between k -nearest neighbor estimator and Bayes regression function, in a very general setting. We also particularize our results to classical function spaces such as Sobolev spaces, Besov spaces and reproducing kernel Hilbert spaces. The rates obtained are genuine nonparametric convergence rates, and up to our knowledge the first of their kind for k –nearest neighbor regression [13] , [25] .