Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Dhruv Shah, Jorge Cortes

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.

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

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.

Paper Structure

This paper contains 19 sections, 8 theorems, 34 equations, 1 figure, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Statement of Contributions
  4. Preliminaries
  5. The Koopman Operator
  6. EDMD as an Orthogonal Projection
  7. Principal Angles and Vectors
  8. Invariance Proximity
  9. Problem Statement
  10. Subspace Pruning via Principal Vectors
  11. Computation of Principal Angles and Vectors in $L_2(\mu_X)$
  12. Single-Principal-Vector (SPV) Pruning
  13. Equivalence of RFB-EDMD and SPV Algorithms
  14. Efficient Computation of Principal Angles and Vectors via Rank-One Updates
  15. Computation of Principal Arguments
  16. ...and 4 more sections

Key Result

Theorem B.3

Let $\mathcal{U},\mathcal{V}\subset\mathbb{R}^n$ have orthonormal basis matrices $Q_{\mathcal{U}}\in\mathbb{R}^{n\times d_1}$ and $Q_{\mathcal{V}}\in\mathbb{R}^{n\times d_2}$, and set $k=\min\{d_1,d_2\}$. Compute the compact SVD where $\sigma_1\ge\cdots\ge\sigma_k\ge0$. Then, the principal angles $\{\theta_j\}_{j=1}^k$ and vectors $\{u_j\}_{j=1}^k$, $\{v_j\}_{j=1}^k$ between $\mathcal{U}$ and $\m

Figures (1)

  • Figure 3: Real part of the leading non-trivial Koopman eigenfunction approximations for the system in \ref{['eq:sys1']} before (left) and after (right) SPV pruning. The pruned eigenfunction is significantly smoother, demonstrating improved approximation quality achieved by the pruning procedure.

Theorems & Definitions (19)

  • Definition B.1: Principal Angles and Vectors
  • Remark B.2
  • Theorem B.3: Computation via SVD AB-GHG:73
  • Definition B.4: Invariance Proximity MH-JC:26-ajc
  • Theorem B.5: Worst-Case Relative Prediction Error MH-JC:26-ajc
  • Proposition D.1
  • proof
  • Lemma D.2
  • proof
  • Theorem D.3: Algorithmic Equivalence
  • ...and 9 more