Efficient Optimization for Linear Dynamical Systems with Applications to Clustering and Sparse Coding

Linear Dynamical Systems (LDSs) are fundamental tools for modeling spatio-temporal data in various disciplines. Though rich in modeling, analyzing LDSs is not free of difficulty, mainly because LDSs do not comply with Euclidean geometry and hence conventional learning techniques can not be applied directly. In this paper, we propose an efficient projected gradient descent method to minimize a general form of a loss function and demonstrate how clustering and sparse coding with LDSs can be solved by the proposed method efficiently. To this end, we first derive a novel canonical form for representing the parameters of an LDS, and then show how gradient-descent updates through the projection on the space of LDSs can be achieved dexterously. In contrast to previous studies, our solution avoids any approximation in LDS modeling or during the optimization process. Extensive experiments reveal the superior performance of the proposed method in terms of the convergence and classification accuracy over state-of-the-art techniques.