LJKProbability & Statistics Seminar

On Thursday June 30 2016 at 14h00 in Amphitheater  RDC  IRMA Tower

Seminary of Clément MARTEAU (Université de Lyon 1)

Classification with the nearest neighbor rule in general finite dimensional spaces: necessary and sufficient conditions

Summary

Given an nsample of random vectors (Xi,Yi)1≤i≤n whose joint law is unknown, the longstanding problem of supervised classification aims to \\textit{optimally} predict the label Y of a given a new observation X. In this context, the nearest neighbor rule is a popular flexible and intuitive method in nonparametric situations.
Even if this algorithm is commonly used in the machine learning and statistics communities, less is known about its prediction ability in general finite dimensional spaces, especially when the support of the density of the observations is R^d. This paper is devoted to the study of the statistical properties of the nearest neighbor rule in various situations. In particular, attention is paid to the marginal law of X, as well as the smoothness and margin properties of the regression function \\eta(X)=\\E[YX]. We identify two necessary and sufficient conditions to obtain uniform consistency rates of classification and to derive sharp estimates in the case of the nearest neighbor rule. Some numerical experiments are proposed at the end of the paper to help illustrate the discussion.
