Learning linear dynamical systems under convex constraints
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We consider the problem of identification of linear dynamical systems from a single trajectory. Recent results have predominantly focused on the setup where no structural assumption is made on the system matrix A^* ∈ℝ^n × n, and have consequently analyzed the ordinary least squares (OLS) estimator in detail. We assume prior structural information on A^* is available, which can be captured in the form of a convex set 𝒦 containing A^*. For the solution of the ensuing constrained least squares estimator, we derive non-asymptotic error bounds in the Frobenius norm which depend on the local size of the tangent cone of 𝒦 at A^*. To illustrate the usefulness of this result, we instantiate it for the settings where, (i) 𝒦 is a d dimensional subspace of ℝ^n × n, or (ii) A^* is k-sparse and 𝒦 is a suitably scaled ℓ_1 ball. In the regimes where d, k ≪ n^2, our bounds improve upon those obtained from the OLS estimator.
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