Approximate Inference and Learning of State Space Models with Laplace Noise

State space models have been extensively applied to model and control dynamical systems in disciplines including neuroscience, target tracking, and audio processing. A common modeling assumption is that both the state and data noise are Gaussian because it simplifies the estimation of the system's state and model parameters. However, in many real-world scenarios where the noise is heavy-tailed or includes outliers, this assumption does not hold, and the performance of the model degrades. In this paper, we present a new approximate inference algorithm for state space models with Laplace-distributed multivariate data that is robust to a wide range of non-Gaussian noise. Exact inference is combined with an expectation propagation algorithm, leading to filtering and smoothing that outperforms existing approximate inference methods for Laplace-distributed data, while retaining a fast speed similar to the Kalman filter. Further, we present a maximum posterior expectation-maximization (EM) algorithm that learns the parameters of the model in an unsupervised way, automatically avoids over-fitting the data, and provides better model estimation than existing methods for the Gaussian model. The quality of the inference and learning algorithms are exemplified through a diverse set of experiments and an application to non-linear tracking of audio frequency.