Probabilistic Kernel Matrix Learning with a Mixture Model of Kernels
This paper addresses the kernel matrix learning problem in kernel methods. We model the kernel matrix as a random positive definite matrix following the Wishart distribution, with the parameter matrix of the Wishart distribution represented as a linear combination of mutually independent matrices with their own Wishart distributions. This defines a probabilistic mixture model of kernels that can be represented as a hierarchical model involving three levels, relating the target kernel matrix, the parameter kernel matrix, and the hyperparameter kernel matrices of the mixture components. Since in most cases a linear combination of Wishart matrices is no longer Wishart, we propose an approximation method by employing a new Wishart matrix to approximate the parameter matrix through preserving the first and second moments of the linear combination of Wishart matrices. Based on the Wishart matrix estimated, kernel matrix learning is then solved by an expectation-maximization (EM) learning algorithm to infer the missing data of the kernel matrix and the unknown parameter matrix of the distribution. Moreover, we study the kernel matrix learning problem in the context of classification problems with the use of a kernel nearest neighbor classifier. Classification experiments on several benchmark data sets show promising results. Furthermore, our method has opened up a possible direction for addressing the kernel