Koopman Subspace Pruning in Reproducing Kernel Hilbert Spaces via Principal Vectors

Researchers Dhruv Shah and Jorge Cortes have published a new paper introducing algorithms for 'Koopman subspace pruning' within Reproducing Kernel Hilbert Spaces (RKHS). The core problem they address is that data-driven approximations of the infinite-dimensional Koopman operator—a powerful tool for analyzing nonlinear dynamical systems—rely on projecting data onto a finite-dimensional subspace. The accuracy of the resulting model depends heavily on how 'invariant' this subspace is under the operator's action. Subspace pruning improves this by systematically discarding data directions that are geometrically misaligned, a misalignment measured by the largest principal angle between the subspace and its transformed image.

The key innovation is that existing pruning techniques were largely confined to Euclidean geometry. Shah and Cortes bridge this gap by formulating how to compute principal angles and vectors within the more complex, high-dimensional RKHS geometry, which is common in kernel-based machine learning. They outline an exact computational method and then scale it for practical use with large datasets using randomized Nyström approximations. Based on this, they introduce two specific algorithms: Kernel-SPV (Subspace Pruning via Principal Vectors) and its approximate counterpart, Approximate Kernel-SPV, for targeted subspace refinement. Simulation results validate that their approach successfully enhances model invariance and, by extension, predictive performance in a kernelized setting.