  As enjoying the closed form solution, least squares support vector machine
(LSSVM) has been widely used for classification and regression problems having
the comparable performance with other types of SVMs. However, LSSVM has two
drawbacks: sensitive to outliers and lacking sparseness. Robust LSSVM (R-LSSVM)
overcomes the first partly via nonconvex truncated loss function, but the
current algorithms for R-LSSVM with the dense solution are faced with the
second drawback and are inefficient for training large-scale problems. In this
paper, we interpret the robustness of R-LSSVM from a re-weighted viewpoint and
give a primal R-LSSVM by the representer theorem. The new model may have sparse
solution if the corresponding kernel matrix has low rank. Then approximating
the kernel matrix by a low-rank matrix and smoothing the loss function by
entropy penalty function, we propose a convergent sparse R-LSSVM (SR-LSSVM)
algorithm to achieve the sparse solution of primal R-LSSVM, which overcomes two
drawbacks of LSSVM simultaneously. The proposed algorithm has lower complexity
than the existing algorithms and is very efficient for training large-scale
problems. Many experimental results illustrate that SR-LSSVM can achieve better
or comparable performance with less training time than related algorithms,
especially for training large scale problems.
