A novel framework for action recognition in video using empirical covariance matrices of bags of low-dimensional feature vectors is developed. The feature vectors are extracted from segments of silhouette tunnels of moving objects and coarsely capture their shapes. The matrix logarithm is used to map the segment covariance matrices, which live in a nonlinear Riemannian manifold, to the vector space of symmetric matrices. A recently developed sparse linear representation framework for dictionary-based classification is then applied to the log-covariance matrices. The log-covariance matrix of a query segment is approximated by a sparse linear combination of the log-covariance matrices of training segments and the sparse coefficients are used to determine the action label of the query segment. This approach is tested on the Weizmann and the UT-Tower human action datasets. The new approach attains a segment-level classification rate of 96.74% for the Weizmann dataset and 96.15% for the UT-Tower dataset. Additionally, the proposed method is computationally and memory efficient and easy to implement.