New Method for Improving Koopman Operator Approximations Using Kernel Hilbert Space Geometry
Researchers have introduced two algorithms, Kernel-SPV and Approximate Kernel-SPV, that extend Koopman subspace pruning techniques from Euclidean settings to Reproducing Kernel Hilbert Spaces (RKHS). The Koopman operator framework is widely used in data-driven dynamical systems modeling, but its accuracy depends on selecting subspaces that are nearly invariant under the operator. The work addresses a significant geometric gap in existing methods and offers a scalable approximation for large datasets.
Data-driven modeling of dynamical systems often relies on finite-dimensional approximations of the infinite-dimensional Koopman operator, whose predictive quality is sensitive to how well the chosen subspace is preserved under the operator's action. Existing subspace pruning methods—which discard geometrically misaligned directions to improve this invariance—have been largely confined to Euclidean spaces. This paper bridges that gap by developing a framework for computing principal angles and vectors within a Reproducing Kernel Hilbert Space geometry. The authors first derive an exact computational routine, then scale it to large datasets using randomized Nyström approximations. Building on these foundations, they propose the Kernel-SPV and Approximate Kernel-SPV algorithms for targeted subspace refinement. Simulation results are reported to validate the approach. The work was submitted to arXiv in April 2026 and revised in June 2026, and is cross-listed under Systems and Control and Machine Learning.
What's missing
The paper reports simulation-based validation only; no empirical benchmarks on real-world dynamical systems datasets are described, leaving open questions about practical performance gains over existing Euclidean pruning methods. The work has not yet undergone formal peer review.
What different sources said
- arXiv stat.MLCenter
Koopman Subspace Pruning in Reproducing Kernel Hilbert Spaces via Principal Vectors
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