Nonparallel Hyperplanes Support Vector Machines

Support vector machine, which is based on the statistical learning theory, is widespread attended for its outstanding learning ability. For the pattern recognition problem, early support vector machine algorithms, such as standard support vector machine, v-support vector machine, least squares support vector machine, have a pair of parallel hyperplane boundaries, which limit the generalization performance. This kind of support vector machines are called parallel hyperplanes support vector machines. However, many non-parallel hyperplanes support vector machines which developed later not only own the advantages of parallel hyperplanes support vector machines but also reflect great superi-ority when dealing with the XOR data. This paper puts forward the following two parallel hyperplanes support vector machines models.1. Least squares support vector machine with parametric margin. The boundaries of least squares support vector machine with parametric margin is no longer parallel, it aims at generating a pair of nonparallel hyperplanes that best across the two classes. There-fore, a better consideration for the classification is the application of multi-hyperplanes. Numerical experiments show the effectiveness of our algorithm.2. The concave-convex procedure of the twin support vector machine. Based on the twin support vector machine, we use the Ramp loss function instead of the hinge loss func-tion, and obtain the optimal value by the concave-convex procedure. Our CCCP-TWSVM inherits the merit of nonparallel multi-hyperplanes decision method, and decreased the number of support vectors, thus improves the sparsity and running speed. In toy datasets and UCI benchmark datasets, our numerical experiments show the feasibility and effec-tiveness of our algorithm.