  Linear Dynamical System (LDS) is an elegant mathematical framework for
modeling and learning multivariate time series. However, in general, it is
difficult to set the dimension of its hidden state space. A small number of
hidden states may not be able to model the complexities of a time series, while
a large number of hidden states can lead to overfitting. In this paper, we
study methods that impose an $\ell_1$ regularization on the transition matrix
of an LDS model to alleviate the problem of choosing the optimal number of
hidden states. We incorporate a generalized gradient descent method into the
Maximum a Posteriori (MAP) framework and use Expectation Maximization (EM) to
iteratively achieve sparsity on the transition matrix of an LDS model. We show
that our Sparse Linear Dynamical System (SLDS) improves the predictive
performance when compared to ordinary LDS on a multivariate clinical time
series dataset.
