Asymptotic predictions of the finite-sample risk of the k-nearest-neighbor classifier

The finite-sample risk of the k-nearest-neighbor classifier is analyzed for a family of two-class problems in which patterns are randomly generated from smooth probability distributions in an n-dimensional Euclidean feature space. First, an exact integral expression for the m-sample risk is obtained for a k-nearest-neighbor classifier that uses a reference sample of m labeled feature vectors. Using a multidimensional application of Laplace's method of integration, this integral can be represented as an asymptotic expansion in negative rational powers of m. The leading terms of this asymptotic expansion elucidate the curse of dimensionality and other properties of the finite-sample risk.