Concave-Convex Programming for Ramp Loss-Based Maximum Margin and Minimum Volume Twin Spheres Machine

Twin hyper-sphere support vector machine (THSVM) classifies two classes of samples via two hyper-spheres instead of a pair of nonparallel hyper-planes in the conventional twin support vector machine. It avoids the matrix inverse operation. However, the THSVM uses hinge loss which easily leads to sensitivity of the noises. In this paper, we propose a ramp loss-based maximum margin and minimum volume twin spheres machine ((RMTSM)-T-3) to enhance the ability of noise resistance. (RMTSM)-T-3 is robust because of introducing the ramp loss function, but it is non-convex. (RMTSM)-T-3 has several admirable advantages. For example, it can explicitly incorporate noises in the training process. In addition, the ramp loss function can be decomposed into the difference of a convex hinge loss and another convex loss. Then the concave-convex programming procedure is employed to efficiently solve the non-convex problems of (RMTSM)-T-3 by solving a sequence of convex programs iteratively. Experimental results on one artificial data set, 14 imbalanced binary datasets and 3 multiclass datasets indicate that our proposed (RMTSM)-T-3 yields a good generalization performance.