Discriminative Analysis for Symmetric Positive Definite Matrices on Lie Groups

In this paper, we study discriminative analysis of symmetric positive definite (SPD) matrices on Lie groups (LGs), namely, transforming an LG into a dimension-reduced one by optimizing data separability. In particular, we take the space of SPD matrices, e.g., covariance matrices, as a concrete example of LGs, which has proved to be a powerful tool for high-order image feature representation. The discriminative transformation of an LG is achieved by optimizing the within-class compactness as well as the between-class separability based on the popular graph embedding framework. A new kernel based on the geodesic distance between two samples in the dimension-reduced LG is then defined and fed into classical kernel-based classifiers, e.g., support vector machine, for various visual classification tasks. Extensive experiments on five public datasets, i.e., Scene-15, Caltech101, UIUC-Sport, MIT-Indoor, and VOC07, well demonstrate the effectiveness of discriminative analysis for SPD matrices on LGs, and the state-of-the-art performances are reported.