Hyperellipsoidal Statistical Classifications in a Reproducing Kernel Hilbert Space

Standard support vector machines (SVMs) have kernels based on the Euclidean distance. This brief extends standard SVMs to SVMs with kernels based on the Mahalanobis distance. The extended SVMs become a special case of the Euclidean distance when the covariance matrix in a reproducing kernel Hilbert space is degenerated to an identity. The Mahalanobis distance leads to hyperellipsoidal kernels and the Euclidean distance results in hyperspherical ones. In this brief, the Mahalanobis distance-based kernel in a reproducing kernel Hilbert space is developed systematically. Extensive experiments demonstrate that the hyperellipsoidal kernels slightly outperform the hyperspherical ones, with fewer SVs.