Robust projection twin extreme learning machines with capped L1-norm distance metric

In this paper, we incorporate the idea of projection twin support vector machines (PTSVM) into the basic framework of twin extreme learning machines (TELM) and first propose a novel binary classifier named projection twin extreme learning machines (PTELM). PTELM is to seek two projection directions in the TELM feature space, such that the projected samples of one class are well separated from those of the other class. Compared with the PTSVM, PTELM tackles nonlinear cases without using several fixed kernel functions, thus PTELM is less sensitive to use specified parameters and can get better classification accuracy. Then, a new capped L1-norm PTELM (CL1-PTELM) is proposed by introducing capped L1-norm distance metric in PTELM to reduce the effect of outliers. CL1-PTELM overcomes the disadvantages of L2-norm distance metric and hinge loss. Thus, CL1-PTELM enhances the robust performance of our PTELM. Finally, two effective algorithms are designed to solve the problem of PTELM and to deal with the challenge of CL1-PTELM brought by non-convex optimization problem, respectively. Simultaneously, we theoretically prove the convergence and local optimality of CL1-PTELM algorithm. Numerical experiments on three synthetic datasets and several UCI datasets show the feasibility and effectiveness of our proposed methods.