Smooth and semi-smooth pinball twin support vector machine

In this paper, we firstly propose a new binary classifier termed as smooth pinball twin support vector machine (SP-TSVM) based on a smooth and everywhere differentiable L2-norm pinball loss. It is closely related to the quantile distance, making the model more robust. The SP-TSVM is less sensitive to feature noise, especially noise located near the decision hyperplane. It not only follows the maximal margin principle, but also avoids the matrix inverse operation. Secondly, since SP-TSVM does not have sparsity, we further propose a semi -smooth L2-norm pinball loss function and establish the model: semi-smooth pinball twin support vector machine (SSP-TSVM). It not only has sparsity and inherits the advantages of SP-TSVM, but also can suppress the negative effects of outliers. Since the SSP-TSVM model is a non-convex optimization problem, this paper adopts the convenient and easy-to-use concave-convex procedure (CCCP) optimization method to solve it. In each step of the iterative process, the model solves a series of SP-TSVM-like problems. Experimental results on 19 data sets indicate the validity of our proposed models.