Koopman Subspace Pruning in Reproducing Kernel Hilbert Spaces via Principal Vectors
This work addresses a domain-specific problem for researchers in dynamical systems and kernel methods, offering an incremental extension of existing techniques to a new geometric setting.
The paper tackles the problem of improving predictive accuracy in data-driven approximations of the Koopman operator by extending subspace pruning techniques from Euclidean settings to Reproducing Kernel Hilbert Spaces (RKHS), and it presents algorithms that are validated through simulation results.
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.