A New Approach to Learning Linear Dynamical Systems

TL;DR

This paper introduces the first polynomial-time algorithm for learning linear dynamical systems from polynomial-length trajectories, achieving near-optimal accuracy under minimal assumptions like observability and controllability.

Contribution

It presents a novel method of moments estimator to directly recover system dynamics from measurements, with proven statistical lower bounds when assumptions are violated.

Findings

First polynomial-time algorithm for learning LDS from polynomial data

Achieves polynomial error bounds under minimal assumptions

Provides statistical lower bounds for cases violating assumptions

Abstract

Linear dynamical systems are the foundational statistical model upon which control theory is built. Both the celebrated Kalman filter and the linear quadratic regulator require knowledge of the system dynamics to provide analytic guarantees. Naturally, learning the dynamics of a linear dynamical system from linear measurements has been intensively studied since Rudolph Kalman's pioneering work in the 1960's. Towards these ends, we provide the first polynomial time algorithm for learning a linear dynamical system from a polynomial length trajectory up to polynomial error in the system parameters under essentially minimal assumptions: observability, controllability, and marginal stability. Our algorithm is built on a method of moments estimator to directly estimate Markov parameters from which the dynamics can be extracted. Furthermore, we provide statistical lower bounds when our observability and controllability assumptions are violated.