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

• URL: http://arxiv.org/abs/2011.10006v3
• Date: Mon, 7 Mar 2022 18:15:21 GMT
• Title: Improved rates for prediction and identification of partially observed linear dynamical systems
• Authors: Holden Lee
• Abstract summary: Identification of a linear time-in dynamical system from partial observations is a fundamental problem in control theory. We propose an algorithm that learns such systems with non-asymptotic statistical rates depending on the inherentity $d$ of the system. Our algorithm is based on multi-scale low-rank approximation SVD applied to Hankel matrices of increasing sizes.
• Score: 4.68299658663016
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $\widetilde O\left(\sqrt\frac{d}{T}\right)$ in $\mathcal{H}_2$ 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 $\mathcal H_\infty$ norm estimation, which may be of independent interest.

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