We consider the problem of joint learning of multiple linear dynamical systems. This has received significant attention recently under different types of assumptions on the model parameters. The setting we consider involves a collection of $m$ linear systems each of which resides on a node of a given undirected graph $G = ([m], \mathcal{E})$. We assume that the system matrices are marginally stable, and satisfy a smoothness constraint w.r.t $G$ -- akin to the quadratic variation of a signal on a graph. Given access to the states of the nodes over $T$ time points, we then propose two estimators for joint estimation of the system matrices, along with non-asymptotic error bounds on the mean-squared error (MSE). In particular, we show conditions under which the MSE converges to zero as $m$ increases, typically polynomially fast w.r.t $m$. The results hold under mild (i.e., $T \sim \log m$), or sometimes, even no assumption on $T$ (i.e. $T \geq 2$).

翻译：我们考虑多个线性动力系统的联合学习问题。近年来，该问题在不同类型的模型参数假设下受到了广泛关注。我们所考虑的设定涉及一个包含 $m$ 个线性系统的集合，其中每个系统位于给定无向图 $G = ([m], \mathcal{E})$ 的一个节点上。我们假设系统矩阵是边缘稳定的，并且满足关于 $G$ 的光滑性约束——类似于图上信号的二次变差。在给定节点状态在 $T$ 个时间点上的观测数据后，我们提出了两个用于联合估计系统矩阵的估计器，并给出了均方误差（MSE）的非渐近误差界。特别地，我们展示了在何种条件下，当 $m$ 增加时，MSE 会收敛到零，通常相对于 $m$ 呈多项式速度收敛。这些结果在温和的（即 $T \sim \log m$）甚至有时对 $T$ 没有假设（即 $T \geq 2$）的条件下成立。