Case StudiesIndefinite kernels in least squares support vector machines and principal component analysis

•LS-SVM with an indefinite kernel is proposed.•kPCA with an indefinite kernel is proposed.•Feature space interpretation for both indefinite LS-SVM and indefinite kPCA is given.•LS-SVM with indefinite kernels for classification and kPCA shows good performance on numerical experiments.

Because of several successful applications, indefinite kernels have attracted many research interests in recent years. This paper addresses indefinite learning in the framework of least squares support vector machines (LS-SVM). Unlike existing indefinite kernel learning methods, which usually involve non-convex problems, the indefinite LS-SVM is still easy to solve, but the kernel trick and primal-dual relationship for LS-SVM with a Mercer kernel is no longer valid. In this paper, we give a feature space interpretation for indefinite LS-SVM. In the same framework, kernel principal component analysis with an infinite kernel is discussed as well. In numerical experiments, LS-SVM with indefinite kernels for classification and kernel principal component analysis is evaluated. Its good performance together with the feature space interpretation given in this paper imply the potential use of indefinite LS-SVM in real applications.