We consider the problem of joint estimation of the parameters of $m$ linear dynamical systems, given access to single realizations of their respective trajectories, each of length $T$. The linear systems are assumed to reside on the nodes of an undirected and connected graph $G = ([m], \mathcal{E})$, and the system matrices are assumed to either vary smoothly or exhibit small number of ``jumps'' across the edges. We consider a total variation penalized least-squares estimator and derive non-asymptotic bounds on the mean squared error (MSE) which hold with high probability. In particular, the bounds imply for certain choices of well connected $G$ that the MSE goes to zero as $m$ increases, even when $T$ is constant. The theoretical results are supported by extensive experiments on synthetic and real data.

翻译：我们考虑在给定各自轨迹的单一实现（每条轨迹长度为 $T$）的情况下，联合估计 $m$ 个线性动力系统参数的问题。假设这些线性系统位于一个无向连通图 $G = ([m], \mathcal{E})$ 的节点上，且系统矩阵在图的边之间要么平滑变化，要么呈现少量“跳跃”。我们研究了一种总变差惩罚最小二乘估计器，并以高概率推导了均方误差（MSE）的非渐近界。特别地，对于某些连通性良好的图 $G$，该界表明当 $m$ 增加时，即使 $T$ 为常数，MSE 也会趋近于零。理论结果得到了在合成数据和真实数据上进行的大量实验的支持。