Twin minimax probability machine for pattern classification

We propose a new distribution-free Bayes optimal classifier, called the twin minimax probability machine (TWMPM), which combines the benefits of both minimax probability machine(MPM) and twin support vector machine (TWSVM). TWMPM tries to construct two nonparallel hyperplanes such that each hyperplane separates one class samples with maximal probability, and is distant from the other class samples simultaneously. Moreover, the proposed TWMPM can control the misclassification error of samples in a worst-case setting by minimizing the upper bound on misclassification probability. An efficient algorithm for TWMPM is first proposed, which transforms TWMPM into concave fractional programming by applying multivariate Chebyshev inequality. Then the proposed TWMPM is reformulated as a pair of convex quadric programming (QP) by proper mathematical transformations. This guarantees TWMPM to have global optimal solution and be solved simply and effectively. In addition, we develop also an iterative algorithm for the proposed TWMPM. By comparing the two proposed algorithms theoretically, it is easy to know that the convex quadric programming algorithm is with lower computation burden than iterative algorithm for the TWMPM. A linear TWMPM version is first built, and then we show how to exploit mercer kernel to obtain nonlinear TWMPM version. The computation complexity for QP algorithm of TWMPM is in the same order as the traditional twin support vector machine (TWSVM). Experiments are carried out on three databases: UCI benchmark database, a practical application database and an artificial database. With low computation complexity and fewer parameters, experiments show the feasibility and effectiveness of the proposed TWMPM and its QP algorithm.