Unsupervised and Semi-supervised Lagrangian Support Vector Machines with Polyhedral Perturbations

Support Vector Machines (SVMs) have been dominant learning techniques for more than ten years, and mostly applied to supervised learning problems. These years two-class unsupervised and semi-supervised classification algorithms based on Bounded $C$-SVMs, Bounded $\nu$-SVMs and Lagrangian SVMs (LSVMs) respectively, which are relaxed to Semi-definite Programming (SDP), get good classification results. These support vector methods implicitly assume that training data in the optimization problems to be known exactly. But in practice, the training data are usually subjected to measurement noise. Zhao et al proposed robust version to Bounded $C-$ SVMs, Bounded $\nu$-SVMs and Lagrangian SVMs (LSVMs) respectively with perturbations in convex polyhedrons and ellipsoids. The region of perturbation in the methods mentioned above is not general, and there are many perturbations in non-convex regions in practice. Therefore we proposed unsupervised and semi-supervised classification problems based on Lagrangian Support Vector Machines with general polyhedral perturbations. But the problem has difficulty to compute, we will find its semi-definite relaxation that can approximate it well. Numerical results confirm the robustness of the proposed method.