Multiple variable splitting is a general technique for decomposing problems by using copies of variables and additional linking constraints that equate their values. The resulting large optimization problem can be solved with a specialized interior-point method that exploits the problem structure and computes the Newton direction with a combination of direct and iterative solvers (i.e., Cholesky factorizations and preconditioned conjugate gradients for linear systems related to, respectively, subproblems and new linking constraints). The present work applies this method to solving real-world binary classification and novelty (or outlier) detection problems by means of, respectively, two-class and one-class linear support vector machines (SVMs). Unlike previous interior-point approaches for SVMs, which were practical only with low-dimensional points, the new proposal can also deal with high-dimensional data. The new method is compared with state-of-the-art solvers for SVMs, that are based on either interior-point algorithms (such as SVM-OOPS) or specific algorithms developed by the machine learning community (such as LIBSVM and LIBLINEAR). The computational results show that, for two-class SVMs, the new proposal is competitive not only against previous interior-point methods---and much more efficient than they are with high-dimensional data---but also against LIBSVM; whereas LIBLINEAR generally outperformed the proposal. For one-class SVMs, the new method consistently outperformed all other approaches, in terms of either solution time or solution quality.
Jordi Castro, New interior-point approach for one- and two-class linear support vector machines using multiple variable splitting, Research Report UPC-DEIO-JC 2021-01, Dept. of Statistics and Operations Research, Universitat Politècnica de Catalunya, Barcelona, Catalonia, 2021.