SVM Training: Second-Order Cone Programming versus Quadratic Programming

The support vector machine (SVM) problem is a convex quadratic programming problem which scales with the training data size. If the training size is large, the problem cannot be solved by straighforward methods. The large-scale SVM problems are tackled by applying chunking (decomposition) technique. The quadratic programming problem involves a square matrix which is called kernel matrix is positive semi-definite. That is, the rank of the kernel matrix is less than or equal to its size. In this paper we discuss a method that can exploit the low-rank of the kernel matrix, and an interior-point method (IPM) is efficiently applied to the global (large-sized) problem. The method is based on the technique of second-order cone programming (SOCP). This method reformulates the SVM's quadratic programming problem into the second-order cone programming problem. The SOCP method is much faster than efficient softwares SVM^lightand SVMTorch if the rank of the kernel matrix is small enough compared to the training set size or if the kernel matrix can be approximated by a low-rank positive semidefinite matrix.