Data in the sciences frequently 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 latent state and observation at each time slice as vectors. We present the multilinear dynamical system (MLDS) for modeling tensor time series and an expectation–maximization (EM) algorithm to estimate the parameters. The MLDS models each tensor observation in a series as the multilinear projection of a corresponding member of a sequence of latent tensors. The latent tensors are again evolving with respect to a multilinear projection. Compared to the LDS with an equal number of parameters, the MLDS achieves higher prediction accuracy and marginal likelihood.
Mark Rogers, Lei Li and Stuart J. Russell (2013), "Multilinear Dynamical Systems for Tensor Time Series", In Advances in Neural Information Processing Systems 26. [PDF]