Joint Learning of Linear Dynamical Systems under Smoothness Constraints

2024-06-03

arXiv.org

AI Summary

This paper addresses the joint identification of multiple linear dynamical systems defined over a graph: each node hosts an marginally stable linear system, and the associated system matrices satisfy a graph-smoothness constraint—formulated analogously to the quadratic variation of graph signals. We propose the first framework that incorporates graph regularization into multi-system joint estimation. Leveraging non-asymptotic statistical analysis and matrix perturbation theory, we derive a tight upper bound on the mean-squared error (MSE). Theoretically, we establish that the MSE decays polynomially to zero in the number of systems $m$, even when trajectory length is extremely short—specifically, $T geq 2$ or $T sim log m$. This result breaks the conventional reliance on long trajectories for consistent identification, ensuring statistical consistency under ultra-short data regimes and significantly enhancing joint learning performance in few-shot settings.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Intelligent Robots: State Estimation
• Multiagent Systems: Multiagent Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Security and Privacy: Large-scale security measurements

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. $T geq 2$).

Problem

• Joint learning of multiple linear dynamical systems
• Estimating system matrices under graph smoothness constraints
• Non-asymptotic error bounds for mean-squared error convergence

Innovation

• Joint learning of multiple linear dynamical systems
• Smoothness constraints on undirected graph
• Non-asymptotic error bounds for MSE