Multilinear Dynamical Systems for Tensor Time Series

Many scientific data occur as sequences of multidimensional arrays called tensors. How can hidden, evolving trends in such data be extracted while preserving the tensor structure? The model that is traditionally used is the linear dynamical system (LDS), which treats the observation at each time slice as a vector. In this paper, we propose the multilinear dynamical system (MLDS) for modeling tensor time series and an expectation-maximization (EM) algorithm to estimate the parameters. The MLDS models each time slice of the tensor time series as the multilinear projection of a corresponding member of a sequence of latent, low-dimensional tensors. Compared to the LDS with an equal number of parameters, the MLDS achieves higher prediction accuracy and marginal likelihood for both simulated and real datasets.