Learning Discriminative Stein Kernel for SPD Matrices and Its Applications

Stein kernel (SK) has recently shown promising performance on classifying images represented by symmetric positive definite (SPD) matrices. It evaluates the similarity between two SPD matrices through their eigenvalues. In this paper, we argue that directly using the original eigenvalues may be problematic because: 1) eigenvalue estimation becomes biased when the number of samples is inadequate, which may lead to unreliable kernel evaluation, and 2) more importantly, eigenvalues reflect only the property of an individual SPD matrix. They are not necessarily optimal for computing SK when the goal is to discriminate different classes of SPD matrices. To address the two issues, we propose a discriminative SK (DSK), in which an extra parameter vector is defined to adjust the eigenvalues of input SPD matrices. The optimal parameter values are sought by optimizing a proxy of classification performance. To show the generality of the proposed method, three kernel learning criteria that are commonly used in the literature are employed as a proxy. A comprehensive experimental study is conducted on a variety of image classification tasks to compare the proposed DSK with the original SK and other methods for evaluating the similarity between SPD matrices. The results demonstrate that the DSK can attain greater discrimination and better align with classification tasks by altering the eigenvalues. This makes it produce higher classification performance than the original SK and other commonly used methods.