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