Koopman Subspace Pruning in Reproducing Kernel Hilbert Spaces via Principal Vectors

TL;DR

This paper introduces a novel method for Koopman subspace pruning in RKHS, leveraging principal vectors and angles to improve model invariance, scalable with randomized Nystrom approximations.

Contribution

It develops an exact and scalable computational routine for principal angles and vectors in RKHS, enabling effective Koopman subspace pruning.

Findings

Simulation results validate the effectiveness of the proposed approach.

The method improves the invariance of Koopman subspace approximations.

Scalable algorithms are introduced for large datasets using randomized Nystrom approximations.

Abstract

Data-driven approximations of the infinite-dimensional Koopman operator rely on finite-dimensional projections, where the predictive accuracy of the resulting models hinges heavily on the invariance of the chosen subspace. Subspace pruning systematically discards geometrically misaligned directions to enhance this invariance proximity, which formally corresponds to the largest principal angle between the subspace and its image under the operator. Yet, existing techniques are largely restricted to Euclidean settings. To bridge this gap, this paper presents an approach for computing principal angles and vectors to enable Koopman subspace pruning within a Reproducing Kernel Hilbert Space (RKHS) geometry. We first outline an exact computational routine, which is subsequently scaled for large datasets using randomized Nystrom approximations. Based on these foundations, we introduce the Kernel-SPV and Approximate Kernel-SPV algorithms for targeted subspace refinement via principal vectors. Simulation results validate our approach.