Finding Koopman Invariant Subspaces via Personalized PageRank
For researchers in data-driven dynamical systems, this provides a principled way to automatically discover Koopman-invariant subspaces, addressing a key bottleneck in Koopman approximation.
The paper introduces a method to detect Koopman-invariant subspaces by identifying zero-block structures in EDMD matrices using personalized PageRank, with finite-sample guarantees scaling as O(1/√M). Experiments on four dynamical systems show the method identifies compact, interpretable dictionaries that yield accurate predictions.
Selecting a finite dictionary of observables whose span is Koopman-invariant is a central challenge in data-driven Koopman operator approximation. We address this problem by exploiting zero-block structure in Extended Dynamic Mode Decomposition (EDMD) matrices. We show that any sub-dictionary whose span is Koopman-invariant induces an exact zero block in the EDMD matrix, even for finite data. We then show that such blocks can be detected by applying PageRank to a row-normalized EDMD matrix constructed from a large initial dictionary. The theory extends to approximately invariant subspaces and yields stronger guarantees for personalized PageRank (PPR) when the seed observables lie inside the target block and reach all observables in that block. Combining EDMD concentration bounds with PageRank perturbation theory gives end-to-end detection guarantees with $O(1/\sqrt{M})$ finite-sample scaling and explicit constants. More generally, without assuming an invariant subspace exists, high PPR mass on a sub-dictionary controls discounted multi-step leakage from the seed observables. Numerical experiments on the Duffing oscillator, Van der Pol oscillator, Lorenz system, and a three-well Ramachandran potential suggest that the method identifies compact, interpretable dictionaries with accurate predictions.