An Optimal Global Nearest Neighbor Metric

A quadratic metric d
_AO
(X, Y) =[(X - Y)
^T
A
_O
(X - Y)]
^1/2
is proposed which minimizes the mean-squared error between the nearest neighbor asymptotic risk and the finite sample risk. Under linearity assumptions, a heuristic argument is given which indicates that this metric produces lower mean-squared error than the Euclidean metric. A nonparametric estimate of Ao is developed. If samples appear to come from a Gaussian mixture, an alternative, parametrically directed distance measure is suggested for nearness decisions within a limited region of space. Examples of some two-class Gaussian mixture distributions are included.