Learning Linear Dynamics from Bilinear Observations
This addresses the challenge of system identification in partially observed dynamical systems with bilinear measurements, which is incremental but provides rigorous theoretical guarantees.
The paper tackles the problem of learning linear dynamical systems from bilinear observations, providing finite-time analysis with statistical error bounds and sample complexity guarantees for learning the unknown dynamics matrices up to a similarity transform.
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.