On the Finite Sample Performance of the Nearest Neighbor Classifier

Abstract

The finite sample performance of a nearest neighbor classifier is analyzed for a two-class pattern recognition problem. An exact integral expression is derived for the m-sample risk R/sub m/ given that a reference m-sample of labeled points, drawn independently from Euclidean n-space according to a fixed probability distribution, is available to the classifier. For a family of smooth distributions characterized by asymptotic expansions in general form, it is shown that the m-sample risk R/sub m/ has a complete asymptotic series expansion R/sub m/ ~ R/spl infin/ + /spl Sigma//sup /spl infin///sub k=1/ ckm/sup -k/n/ (m - /spl infin/) where R/spl infin/ denotes the nearest neighbor risk in the infinite-sample limit. Improvements in convergence rate are shown under stronger smoothness assumptions, and in particular, R/sub m/ = R/spl infin/ + O(m/sup -2/n/) if the class-conditional probability densities have uniformly bounded third derivatives on their probability one support. This analysis thus provides further analytic validation of Bellman's curse of dimensionality. Numerical simulations corroborating the formal results are included, and extensions of the theory discussed. The analysis also contains a novel application of Laplace's asymptotic method of integration to a multidimensional integral where the integrand attains its maximum on a continuum of points.

Locations

• Infoscience (Ecole Polytechnique Fédérale de Lausanne) - View - PDF
• CaltechAUTHORS (California Institute of Technology) - View - PDF