Tensor Sparse Coding for Region Covariances

Sparse representation of signals has been the focus of much research in the recent years. A vast majority of existing algorithms deal with vectors, and higher–order data like images are usually vectorized before processing. However, the structure of the data may be lost in the process, leading to poor representation and overall performance degradation. In this paper we propose a novel approach for sparse representation of positive definite matrices, where vectorization would have destroyed the inherent structure of the data. The sparse decomposition of a positive definite matrix is formulated as a convex optimization problem, which falls under the category of determinant maximization (MAXDET) problems [1], for which efficient interior point algorithms exist. Experimental results are shown with simulated examples as well as in real–world computer vision applications, demonstrating the suitability of the new model. This forms the first step toward extending the cornucopia of sparsity-based algorithms to positive definite matrices.