A Unified Algebraic Framework for Subspace Pruning in Koopman Operator Approximation via Principal Vectors

TL;DR

This paper introduces an algebraic framework for subspace pruning in Koopman operator approximation, improving accuracy and scalability by leveraging principal vectors and efficient numerical updates.

Contribution

It presents a unified algebraic approach to subspace pruning, establishing equivalence with existing methods and enhancing computational efficiency.

Findings

The framework effectively minimizes subspace approximation errors.

The numerical scheme reduces computational complexity significantly.

Numerical simulations demonstrate the approach's effectiveness.

Abstract

Finite-dimensional approximations of the Koopman operator rely critically on identifying nearly invariant subspaces. This invariance proximity can be rigorously quantified via the principal angles between a candidate subspace and its image under the operator. To systematically minimize this error, we propose an algebraic framework for subspace pruning utilizing principal vectors. We establish the equivalence of this approach to existing consistency-based methods while providing a foundation for broader generalizations. To ensure scalability, we introduce an efficient numerical update scheme based on rank-one modifications, reducing the computational complexity of tracking principal angles by an order of magnitude. Finally, we demonstrate the effectiveness of our framework through numerical simulations.