Tensor Decompositions Meet Control Theory: Learning General Mixtures of Linear Dynamical Systems
This addresses the challenge of modeling time-series data with subpopulations for applications in data analysis, though it is incremental as it builds on existing tensor decomposition methods.
The paper tackles the problem of learning mixtures of linear dynamical systems without strong separation conditions, using tensor decompositions to achieve results that compete with Bayes optimal clustering and work in partially-observed settings.
Recently Chen and Poor initiated the study of learning mixtures of linear dynamical systems. While linear dynamical systems already have wide-ranging applications in modeling time-series data, using mixture models can lead to a better fit or even a richer understanding of underlying subpopulations represented in the data. In this work we give a new approach to learning mixtures of linear dynamical systems that is based on tensor decompositions. As a result, our algorithm succeeds without strong separation conditions on the components, and can be used to compete with the Bayes optimal clustering of the trajectories. Moreover our algorithm works in the challenging partially-observed setting. Our starting point is the simple but powerful observation that the classic Ho-Kalman algorithm is a close relative of modern tensor decomposition methods for learning latent variable models. This gives us a playbook for how to extend it to work with more complicated generative models.