Learning Causal Estimates of Linear Operators From Noisy Data

This article studies the identification problem of linear systems from a set of noisy input–output trajectories. The problem is formulated and solved as a least-square regularized estimate on a suitable function space of finite-bandwidth operators. This abstract setting is well suited to represent a broad class of finite- and infinite-dimensional linear systems. We determine the value of the regularization parameter as a function of the amount of noise on the learning trajectories and we show how to obtain recursive and causal estimates for the case of linear dynamical systems.