Sparse coding and dictionary learning with linear dynamical systems

Linear Dynamical Systems (LDSs) are the fundamental tools for encoding spatio-temporal data in various disciplines. To enhance the performance of LDSs, in this paper, we address the challenging issue of performing sparse coding on the space of LDSs, where both data and dictionary atoms are LDSs. Rather than approximate the extended observability with a finite-order matrix, we represent the space of LDSs by an infinite Grassmannian consisting of the orthonormalized extended observability subspaces. Via a homeomorphic mapping, such Grassmannian is embedded into the space of symmetric matrices, where a tractable objective function can be derived for sparse coding. Then, we propose an efficient method to learn the system parameters of the dictionary atoms explicitly, by imposing the symmetric constraint to the transition matrices of the data and dictionary systems. Moreover, we combine the state covariance into the algorithm formulation, thus further promoting the performance of the models with symmetric transition matrices. Comparative experimental evaluations reveal the superior performance of proposed methods on various tasks including video classification and tactile recognition.