Improved rates for prediction and identification of partially observed linear dynamical systems

Identification of a linear time-invariant dynamical system from partial observations is a fundamental problem in control theory. Particularly challenging are systems exhibiting long-term memory. A natural question is how learn such systems with non-asymptotic statistical rates depending on the inherent dimensionality (order) d of the system, rather than on the possibly much larger memory length. We propose an algorithm that given a single trajectory of length T with gaussian observation noise, learns the system with a near-optimal rate of widetilde Oleft(√fracdTright) in H2 error, with only logarithmic, rather than polynomial dependence on memory length. We also give bounds under process noise and improved bounds for learning a realization of the system. Our algorithm is based on multi-scale low-rank approximation: SVD applied to Hankel matrices of geometrically increasing sizes. Our analysis relies on careful application of concentration bounds on the Fourier domain -- we give sharper concentration bounds for sample covariance of correlated inputs and for mathcal H_∞ norm estimation, which may be of independent interest.