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 three 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). 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.

翻译：我们考虑从单条轨迹的 $T$ 个样本中实现线性动力系统的有限时间内辨识问题。近期研究主要集中在系统矩阵 $A^* \in \mathbb{R}^{n \times n}$ 无结构假设的设定下，并就此详细分析了普通最小二乘（OLS）估计量。本文假设关于 $A^*$ 的先验结构信息可用，该信息可表示为包含 $A^*$ 的凸集 $\mathcal{K}$ 形式。针对由此产生的约束最小二乘估计量的解，我们推导出 Frobenius 范数下的非渐近误差界，该误差界依赖于 $\mathcal{K}$ 在 $A^*$ 处的局部尺度。为阐明这些结果的实用性，我们将其应用于三个示例：（i）$A^*$ 为稀疏矩阵且 $\mathcal{K}$ 为适当缩放的 $\ell_1$ 球；（ii）$\mathcal{K}$ 为子空间；（iii）$\mathcal{K}$ 由在均匀 $n \times n$ 网格上采样二元凸函数形成的矩阵构成（凸回归）。在所有情形下，我们证明当 $T$ 远小于无约束设定所需值时，仍可对 $A^*$ 进行可靠估计。