Supervised Data Classification via Max-min Separability

Abstract

The problem of discriminating between the elements of two finite sets of points in n-dimensional space is a fundamental in supervised data classification. In practice, it is unlikely for the two sets to be linearly separable. In this paper we consider the problem of separating of two finite sets of points by means of piecewise linear functions. We prove that if these two sets are disjoint then they can be separated by a piecewise linear function and formulate the problem of finding the latter function as an optimization problem with an objective function containing max-min of linear functions. The differential properties of the objective function are studied and an algorithm for its minimization is developed. We present the results of numerical experiments with real world data sets. These results demonstrate the effectiveness of the proposed algorithm for separating two finite sets of points. They also demonstrate the effectiveness of an algorithm based on the concept of max-min separability for solving supervised data classification problems.