Learning Finite-Dimensional Representations For Koopman Operators

In this work, the problem of learning Koopman operator of a discrete-time autonomous system is considered. The learning problem is formulated as a constrained regularized optimization over the infinite-dimensional space of linear operators. We show that under certain but general conditions, a representer theorem holds for the learning problem. This allows reformulating the problem in a finite-dimensional space without loss of any precision. Following this, we consider various cases of regularization and constraint for the latent Koopman operator including the operator norm, the Frobenius norm, and rank. Subsequently, we derive the corresponding finite-dimensional problem.