🤖 AI Summary
This paper addresses the finite-sample identification of the system matrix (A^*) for linear dynamical systems under convex set constraints, based on a single trajectory of length (T). To overcome the low sample efficiency of conventional unconstrained estimators, we propose a constrained least-squares estimation framework. We establish, for the first time, non-asymptotic error bounds for this estimator, explicitly quantifying how local geometric properties—such as the local Rademacher complexity—affect sample complexity. Our method integrates convex optimization with structured modeling to uniformly handle four canonical structural priors: sparsity, subspace constraints, convex regression, and Lipschitz row-wise constraints. Theoretically, we prove that, under such structural constraints, reliable estimation is achievable with significantly fewer samples than required in the unconstrained setting—thereby substantially improving identification efficiency in the small-sample regime.
📝 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 $A^* in mathbb{R}^{n imes n}$, 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 $n imes n$ 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.
