Optimal sparse kernel learning in the Empirical Kernel Feature Space for hyperspectral classification

In this paper, we present a novel framework for sparse kernel learning in a finite space called the Empirical Kernel Feature Space (EKFS). The EKFS can be explicitly built by using any positive definite kernel including Gaussian RBF kernel via an empirical kernel map. In order to turn the empirical kernel map into a feature map associated with a kernel, EKFS is endowed with the dot product of a map associated with the correponding whitened EKFS. In previous sparse kernel learning techniques, subsets of features were selected from the original input feature space. This method was optimal up to the linear kernel. In this work, feature subset selection is performed in the EKFS which leads to the selection of corresponding Reproducing Kernel Hilbert Space (RKHS). Both the EKFS and the corresponding RKHS have the same geometrical structure. The proposed sparse kernel learning can optimally select multiple subsets of newly mapped features in the EKFS in order to improve the generalization performance of the classifier. The sparse kernel-based learning is tested on several hyperspectral data sets and a performance comparison among different feature selection techniques is presented.