arXiv:2303.15121v4 Announce Type: replace-cross
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 either no structural assumption is made on the system matrix $A^* in mathbb{R}^{n times n}$, or specific structural assumptions (e.g. sparsity) are made on $A^*$. 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.