Optimal Expected-Distance Separating Halfspace

One recently proposed criterion to separate two data sets in discriminant analysis is to use a hyperplane, which minimizes the sum of distances to it from all the misclassified data points. Here all distances are supposed to be measured by way of some fixed norm, while misclassification means lying in the wrong halfspace. In this paper we study the problem of determining such an optimal halfspace when points are distributed according to an arbitrary random vector X in ℝ^d. In the unconstrained case in dimension d, we prove that any optimal separating halfspace always balances the misclassified points. Moreover, under polyhedrality assumptions on the support of X, there always exists an optimal separating halfspace passing through d affinely independent points. These results extend in a natural way when different norms (or a fixed gauge) are used to measure distances, and we allow constraints modeling that certain points are forced to be correctly classified.