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Support Vector Machines (SVMs) are among the most popular and the bestperforming classification algorithms. Various approaches have been proposed toreduce the high computation and memory cost when training and predicting basedon large-scale datasets with kernel SVMs. A popular one is the linearizationframework, which successfully builds a bridge between the $L_1$-loss kernel SVMand the $L_1$-loss linear SVM. For linear SVMs, very recently, a semismoothNewton's method is proposed. It is shown to be very competitive and have lowcomputational cost. Consequently, a natural question is whether it is possibleto develop a fast semismooth Newton's algorithm for kernel SVMs. Motivated bythis question and the idea in linearization framework, in this paper, we focuson the $L_2$-loss kernel SVM and propose a semismooth Newton's method basedlinearization and approximation approach for it. The main idea of this approachis to first set up an equivalent linear SVM, then apply the Nystr\"om method toapproximate the kernel matrix, based on which a reduced linear SVM is obtained.Finally, the fast semismooth Newton's method is employed to solve the reducedlinear SVM. We also provide some theoretical analyses on the approximation ofthe kernel matrix. The advantage of the proposed approach is that it maintainslow computational cost and keeps a fast convergence rate. Results of extensivenumerical experiments verify the efficiency of the proposed approach in termsof both predicting accuracy and speed.

A Semismooth-Newton's-Method-Based Linearization and Approximation Approach for Kernel Support Vector Machines