Maximin Linear Discrimination, I

A solution is given to the linear discrimination problem for more than two statistical classes, using a generalized Fisher criterion as the distance measure. Essentially, we find the direction X on which the projections of k > 2 statistical hypotheses make the generalized Fisher criterion maximum. Since the latter depends mainly on the minimum pairwise projected mean difference, the optimal projection direction X maximizes the worst distance. With the use of linear manifold subspaces and decomposition of the optimization problem into a union of simple convex constrained ones, a closed form solution for the optimal X is attained, and no numerical optimization techniques are needed. Such numerical optimization algorithms in high-dimensional spaces were required in previously proposed methods in which other distance measures were used. For the same generalized Fisher distance measure and with similar methodology, we also derive the best set of discriminant vectors.