Subspace Pruning via Principal Vectors for Accurate Koopman-Based Approximations
For researchers using Koopman-based methods for nonlinear dynamics, this work provides a principled approach to improve approximation accuracy and scalability, though it is an incremental improvement over existing consistency-based methods.
This paper proposes a unified algebraic framework for subspace pruning to refine invariance error in Koopman operator approximations, introducing a hybrid strategy that balances multiple and single principal vector pruning. The method reduces computational complexity by an order of magnitude via rank-one modifications and improves numerical stability.
The accuracy of Koopman operator approximations over finite-dimensional spaces relies critically on their invariance properties. These can be rigorously quantified via the principal angles between a candidate subspace and its image under the Koopman operator. This paper proposes a unified algebraic framework for subspace pruning designed to systematically refine the invariance error. We establish the geometric equivalence between consistency-based methods and principal-vector pruning, and build on this insight to introduce a hybrid strategy that balances between multiple and single principal vector pruning for improved numerical stability and scalability. We derive error bounds for the retention of approximate and external eigenfunctions, demonstrating that the multi-vector approach mitigates the numerical drift inherent to sequential pruning. To ensure scalability, we develop an efficient numerical update scheme based on rank-one modifications that reduces the computational complexity of tracking principal angles by an order of magnitude. Finally, we exploit the subspace obtained from the pruning algorithms to build a lifted linear model for state prediction that accounts for the trade-offs between improving invariance and minimizing state reconstruction error. Simulations demonstrate the effectiveness of our approach.