Learning Linear Dynamics from Bilinear Observations

TL;DR

This paper develops a finite-time analysis method for learning linear dynamical systems with bilinear observations, addressing challenges of heavy-tailed data and dependent inputs, and providing error bounds and sample complexity insights.

Contribution

It introduces a novel analysis framework for learning system dynamics from bilinear observations with dependent, heavy-tailed data, including data-dependent and data-independent error bounds.

Findings

Provides high probability error bounds for fixed inputs.

Derives error bounds for random input designs.

Estimates sample complexity for learning dynamics matrices.

Abstract

We consider the problem of learning a realization of a partially observed dynamical system with linear state transitions and bilinear observations. Under very mild assumptions on the process and measurement noises, we provide a finite time analysis for learning the unknown dynamics matrices (up to a similarity transform). Our analysis involves a regression problem with heavy-tailed and dependent data. Moreover, each row of our design matrix contains a Kronecker product of current input with a history of inputs, making it difficult to guarantee persistence of excitation. We overcome these challenges, first providing a data-dependent high probability error bound for arbitrary but fixed inputs. Then, we derive a data-independent error bound for inputs chosen according to a simple random design. Our main results provide an upper bound on the statistical error rates and sample complexity of learning the unknown dynamics matrices from a single finite trajectory of bilinear observations.