Learning linear dynamical systems under convex constraints
(2303.15121)Abstract
We consider the problem of finite-time identification of linear dynamical systems from $T$ samples of a single trajectory. Recent results have predominantly focused on the setup where no structural assumption is made on the system matrix $A* \in \mathbb{R}{n \times n}$, 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 $\mathcal{K}$ containing $A*$. For the solution of the ensuing constrained least squares estimator, we derive non-asymptotic error bounds in the Frobenius norm that depend on the local size of $\mathcal{K}$ at $A*$. To illustrate the usefulness of these results, we instantiate them for four examples, namely when (i) $A*$ is sparse and $\mathcal{K}$ is a suitably scaled $\ell_1$ ball; (ii) $\mathcal{K}$ is a subspace; (iii) $\mathcal{K}$ consists of matrices each of which is formed by sampling a bivariate convex function on a uniform $n \times n$ grid (convex regression); (iv) $\mathcal{K}$ consists of matrices each row of which is formed by uniform sampling (with step size $1/T$) of a univariate Lipschitz function. In all these situations, we show that $A*$ can be reliably estimated for values of $T$ much smaller than what is needed for the unconstrained setting.
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