Complex Support Vector Machines for Regression and Quaternary Classification

The paper presents a new framework for complex support vector regression (SVR) as well as Support Vector Machines (SVM) for quaternary classification. The method exploits the notion of widely linear estimation to model the input-out relation for complex-valued data and considers two cases: 1) the complex data are split into their real and imaginary parts and a typical real kernel is employed to map the complex data to a complexified feature space and 2) a pure complex kernel is used to directly map the data to the induced complex feature space. The recently developed Wirtinger's calculus on complex reproducing kernel Hilbert spaces is employed to compute the Lagrangian and derive the dual optimization problem. As one of our major results, we prove that any complex SVM/SVR task is equivalent with solving two real SVM/SVR tasks exploiting a specific real kernel, which is generated by the chosen complex kernel. In particular, the case of pure complex kernels leads to the generation of new kernels, which have not been considered before. In the classification case, the proposed framework inherently splits the complex space into four parts. This leads naturally to solving the four class-task (quaternary classification), instead of the typical two classes of the real SVM. In turn, this rationale can be used in a multiclass problem as a split-class scenario based on four classes, as opposed to the one-versus-all method; this can lead to significant computational savings. Experiments demonstrate the effectiveness of the proposed framework for regression and classification tasks that involve complex data.