Joint learning of a network of linear dynamical systems via total variation penalization

TL;DR

This paper introduces a total variation penalized approach for jointly estimating parameters of multiple linear dynamical systems on a graph, achieving consistent estimation as the number of systems grows, even with limited data per system.

Contribution

It proposes a novel total variation penalized estimator for networked linear dynamical systems and provides non-asymptotic error bounds demonstrating its effectiveness.

Findings

MSE decreases as the number of systems increases for well-connected graphs.

The method performs well on synthetic and real data.

Estimates remain accurate with limited trajectory length T.

Abstract

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.