Analyzing Linear Dynamical Systems: From Modeling to Coding and Learning.

TL;DR: The proposed methods yield higher accuracies in various classification tasks including video classification and tactile recognition as well as efficient algorithms to perform sparse coding and dictionary learning on the space of infinite-dimensional subspaces.

Abstract: Encoding time-series with Linear Dynamical Systems (LDSs) leads to rich models with applications ranging from dynamical texture recognition to video segmentation to name a few. In this paper, we propose to represent LDSs with infinite-dimensional subspaces and derive an analytic solution to obtain stable LDSs. We then devise efficient algorithms to perform sparse coding and dictionary learning on the space of infinite-dimensional subspaces. In particular, two solutions are developed to sparsely encode an LDS. In the first method, we map the subspaces into a Reproducing Kernel Hilbert Space (RKHS) and achieve our goal through kernel sparse coding. As for the second solution, we propose to embed the infinite-dimensional subspaces into the space of symmetric matrices and formulate the sparse coding accordingly in the induced space. For dictionary learning, we encode time-series by introducing a novel concept, namely the two-fold LDSs. We then make use of the two-fold LDSs to derive an analytical form for updating atoms of an LDS dictionary, i.e., each atom is an LDS itself. Compared to several baselines and state-of-the-art methods, the proposed methods yield higher accuracies in various classification tasks including video classification and tactile recognition.