Koopman Subspace Pruning in Reproducing Kernel Hilbert Spaces via Principal Vectors

2026-04-01

AI Summary

Existing data-driven Koopman operator methods struggle to ensure approximate invariance of subspaces under the operator in non-Euclidean settings, limiting predictive accuracy. This work addresses this challenge by extending principal vector–guided subspace pruning to reproducing kernel Hilbert spaces (RKHS) for the first time. By precisely computing principal angles and vectors in RKHS, we introduce Kernel-SPV and its computationally efficient Nyström approximation–based variant, Approximate Kernel-SPV. These approaches overcome the limitations of traditional Euclidean formulations, significantly enhancing the invariance of Koopman-invariant subspaces while maintaining scalability and substantially improving prediction accuracy.

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.

Problem

• Koopman operator
• subspace pruning
• Reproducing Kernel Hilbert Space
• principal angles
• data-driven approximation

Innovation

• Koopman operator
• subspace pruning
• Reproducing Kernel Hilbert Space
• principal vectors
• Nyström approximation