Monotonicity of Linear Separability Under Translation

A set of n pattern vectors are given in d-space and classified arbitrarily into two sets. The sets of patterns are said to be linearly separable if there exists a hyperplane that separates them. We ask whether translation of one of these sets in an arbitrary direction helps separability. Sometimes yes and sometimes no, but yes on the average. The average is taken over all classifications of the patterns into two sets. In fact, we prove that the probability of separability increases as the translation increases. Thus, we conclude that if points are drawn equiprobably from densities fo(x) and f1(x) = fo(x + tw) then the probability of linear separability is minimum at t = 0 and increases with t for t > 0.