Nearest neighbor classification in infinite dimension

¹ IRISA /INRIA Campus de Beaulieu 35042 Rennes Cédex, France. Frederic.Cerou@inria.fr
² Université de Rennes 2, Campus de Villejan, 35043 Rennes Cedex, France. Arnaud.Guyader@uhb.fr

Received: 7 February 2006
Revised: 24 March 2006
Revised: 6 April 2006

Abstract

Let X be a random element in a metric space (F,d), and let Y be a random variable with value 0 or 1. Y is called the class, or the label, of X. Let (X_i,Y_i)_{1 ≤ i ≤ n} be an observed i.i.d. sample having the same law as (X,Y). The problem of classification is to predict the label of a new random element X. The k-nearest neighbor classifier is the simple following rule: look at the k nearest neighbors of X in the trial sample and choose 0 or 1 for its label according to the majority vote. When , Stone (1977) proved in 1977 the universal consistency of this classifier: its probability of error converges to the Bayes error, whatever the distribution of (X,Y). We show in this paper that this result is no longer valid in general metric spaces. However, if (F,d) is separable and if some regularity condition is assumed, then the k-nearest neighbor classifier is weakly consistent.