Joint learning of linear dynamical systems under smoothness constraints

Abstract 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 $G = ([m], {\mathscr{E}})$. We assume that the system matrices are marginally stable, and satisfy a smoothness constraint w.r.t $G$—the smoothness measured in a manner 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. $T \geq 2$).