k nearest neighbors in search of a metric

The finite-sample risk of the k-nearest neighbor classifier that uses a weighted L_p metric as a measure of class similarity is examined. For a family of multiclass, classification problems with smooth distributions in Rⁿ, the risk is represented as an asymptotic expansion in decreasing fractional powers of the reference sample size. An analysis of the leading coefficients reveals that the optimal metric (i.e., the metric that minimizes the risk) tends to a weighted Euclidean (i.e., L₂) metric as the sample size is increased. Numerical calculations corroborate this finding