A Study on Sigmoid Kernels for SVM and the Training of non-PSD Kernels by SMO-type Methods

The sigmoid kernel was quite popular for support vector machines due to its origin from neural networks. Although it is known that the kernel matrix may not be positive semi-definite (PSD), other properties are not fully studied. In this paper, we discuss such non-PSD kernels through the viewpoint of separability. Results help to validate the possible use of non-PSD kernels. One example shows that the sigmoid kernel matrix is conditionally positive definite (CPD) in certain parameters and thus are valid kernels there. However, we also explain that the sigmoid kernel is not better than the RBF kernel in general. Experiments are given to illustrate our analysis. Finally, we discuss how to solve the non-convex dual problems by SMO-type decomposition methods. Suitable modifications for any symmetric non-PSD kernel matrices are proposed with convergence proofs.