Joint Learning of Linear Dynamical Systems under Smoothness Constraints

TL;DR

This paper introduces methods for jointly learning multiple linear dynamical systems arranged on a graph, leveraging smoothness constraints to improve estimation accuracy with minimal data, and provides theoretical guarantees on error decay.

Contribution

It proposes two estimators for joint system matrix estimation under graph-based smoothness constraints with non-asymptotic error bounds and conditions for polynomial convergence of MSE.

Findings

MSE converges to zero as the number of systems increases

Error bounds are non-asymptotic and explicit

Convergence can occur with minimal time points, T ≥ 2

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 $G = ([m], \mathcal{E})$. 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. $T \geq 2$).