Abstract
By representing each image set as a nonsingular covariance matrix on the symmetric positive definite (SPD) manifold, visual classification with image sets has attracted much attention. Despite the success made so far, the issue of large within-class variability of representations still remains a key challenge. Recently, several SPD matrix learning methods have been proposed to assuage this problem by directly constructing an embedding mapping from the original SPD manifold to a lower dimensional one. The advantage of this type of approach is that it cannot only implement discriminative feature selection but also preserve the Riemannian geometrical structure of the original data manifold. Inspired by this fact, we propose a simple SPD manifold deep learning network (SymNet) for image set classification in this article. Specifically, we first design SPD matrix mapping layers to map the input SPD matrices into new ones with lower dimensionality. Then, rectifying layers are devised to activate the input matrices for the purpose of forming a valid SPD manifold, chiefly to inject nonlinearity for SPD matrix learning with two nonlinear functions. Afterward, we introduce pooling layers to further compress the input SPD matrices, and the log-map layer is finally exploited to embed the resulting SPD matrices into the tangent space via log-Euclidean Riemannian computing, such that the Euclidean learning applies. For SymNet, the (2-D) 2 principal component analysis (PCA) technique is utilized to learn the multistage connection weights without requiring complicated computations, thus making it be built and trained easier. On the tail of SymNet, the kernel discriminant analysis (KDA) algorithm is coupled with the output vectorized feature representations to perform discriminative subspace learning. Extensive experiments and comparisons with state-of-the-art methods on six typical visual classification tasks demonstrate the feasibility and validity of the proposed SymNet.