Nonlinear Extension of Multiobjective Multiclass Support Vector Machine Based on the One-against-all Method

Recently, some kinds of extensions of the binary support vector machine (SVM) to multiclass classification have been proposed. In this paper, we focus on the multiobjective multiclass support vector machine based on the one-against-all method (MMSVM-OA), which is an improved new model from one-against-all and all-together methods. The model finds a weighted combination of binary SVMs obtained by the one-against-all method whose weights are determined in order to maximize geometric margins of its multiclass discriminant function for the generalization ability similarly to the all-together method. In addition, the model does not require a large amount of computational resources, while it is reported that it outperforms than one-against-all and all-together methods in numerical experiments. However, it is not formulated as a quadratic programming problem unlike to standard SVMs, it is difficult to apply the kernel method to it. Therefore, in this paper, we propose a nonlinear model derived by a transformation of the MMSVM-OA, which the kernel method can apply to, and show the corresponding multiclass classifier is obtained by solving a convex second-order cone programming problem. Moreover, we show the advantage of the proposed model through numerical experiments.