A Primal Framework for Indefinite Kernel Learning

Abstract

Kernel methods have been widely applied in machine learning to solve complex nonlinear problems. Kernel selection is one of the key issues in kernel methods, since it is vital for improving generalization performance. Traditionally, the selection of kernel is restricted to be positive definite which makes their applicability partially limited. Actually, in many real applications such as gene identification and object recognition, indefinite kernels frequently emerge and can achieve better performance. However, compared to positive definite ones, indefinite kernels are more complicated due to the non-convexity of the subsequent optimization problems, which leads to the incapability of most existing kernel algorithms. Some indefinite kernel methods have been proposed based on the dual of support vector machine (SVM), which mostly emphasize on how to transform the non-convex optimization to be convex by using positive definite kernels to approximate indefinite ones. In fact, the duality gap in SVM usually exists in the case of indefinite kernels and therefore these algorithms do not indeed solve the indefinite kernel problems themselves. In this paper, we present a novel framework for indefinite kernel learning derived directly from the primal of SVM, which establishes several new models not only for single indefinite kernel but also extends to multiple indefinite kernel scenarios. Several algorithms are developed to handle the non-convex optimization problems in these models. We further provide a constructive approach for kernel selection in the algorithms by using the theory of similarity functions. Experiments on real world datasets demonstrate the superiority of our models.