Subspace Pruning via Principal Vectors for Accurate Koopman-Based Approximations

Dhruv Shah , Jorge Cort\'es

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Pith reviewed 2026-05-14 18:52 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords pruningprincipalsubspaceerrorinvariancenumericalanglesapproach

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The pith

A hybrid principal-vector pruning framework refines Koopman subspace invariance with error bounds and rank-one update efficiency for lifted linear prediction.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Koopman operators turn nonlinear dynamics into linear ones by lifting the state into a higher-dimensional space. The accuracy of any finite approximation depends on how invariant the chosen subspace is under the operator. This work treats invariance error through principal angles between a subspace and its image. It shows that existing consistency methods are geometrically the same as pruning along principal vectors. From this equivalence it builds a hybrid pruning rule that removes either one or several vectors at once. The hybrid rule comes with error bounds on how well approximate eigenfunctions and external modes are kept. To keep the computation cheap, the method uses rank-one matrix updates so that tracking the angles costs far less than recomputing from scratch. The final pruned subspace is then used to build a lifted linear predictor that trades off invariance quality against reconstruction error. Simulations are said to confirm the approach works on example systems.

Core claim

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.

Load-bearing premise

That the principal angles between a candidate subspace and its image under the Koopman operator provide a sufficient and refinable measure of invariance error that can be systematically reduced by pruning without losing essential dynamical information.

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper links principal-vector pruning to consistency methods in Koopman approximations and adds a hybrid strategy with efficient updates, but its advantages over simpler methods remain to be verified in detail.

read the letter

The punchline here is that the authors give a geometric view of subspace pruning in Koopman approximations by linking it to principal vectors, and they turn that into a hybrid algorithm with a fast update rule. They start from the observation that invariance can be measured by principal angles between a subspace and its image under the Koopman operator. From there they prove an equivalence to consistency methods, which lets them import some ideas and then propose pruning either one vector or several at once depending on the situation. The hybrid choice is meant to keep the numerics stable while still cutting the error. They also supply bounds on how much the approximate eigenfunctions change when you prune, and they show a rank-one modification trick that makes recomputing the angles much cheaper. That efficiency piece is where the paper adds something concrete. Tracking principal angles from scratch each time would be expensive for large subspaces, so the update scheme addresses a real implementation bottleneck. The connection to building the final lifted predictor that balances invariance and state error is also handled explicitly, which is helpful. The main soft spot is the reliance on the principal angles as the right error measure. It is a natural choice from linear algebra, but in practice with finite data or when the Koopman operator is only approximated, those angles might not capture all the relevant dynamical features. The simulations are said to show effectiveness, yet without detailed comparisons or ablation studies visible in the abstract, it's unclear whether the hybrid method outperforms simpler baselines by enough to matter. If the gains are small, the extra machinery may not be worth it for most users. Overall this is targeted at researchers already working with data-driven linear embeddings of nonlinear systems, particularly in control applications where you need the lifted model to be both accurate and stable. It refines rather than replaces current practice. I recommend sending it out for peer review. The ideas are specific, the claims are testable, and the area can use more algorithmic work on subspace selection.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the approach relies on standard properties of principal angles and rank-one updates within existing Koopman operator theory.

pith-pipeline@v0.9.0 · 5517 in / 1103 out tokens · 54313 ms · 2026-05-14T18:52:06.109846+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We establish the geometric equivalence between consistency-based methods and principal-vector pruning... hybrid MPV-SPV strategy... rank-one modifications... invariance proximity δ(S) = sin θ_max(S,KS)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

principal angles... SVD computation... external eigenfunction retention bounds

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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Construct the Re-alignment Matrix LetE∈R (s−k)×(s−k) be the matrix of eigenvectors. We construct a transformation matrixT∈R s×(s−k) by padding Ewith zeros to align with the originals-dimensional space: T= E 0k×(s−k) .(25)
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Update the Triangular Factor We apply the transformationTto the existing upper triangular factorRto form the intermediate matrixC=RT∈R s×(s−k). We then perform a QR decomposition ofCas C=Q CRC,(26) whereQ C ∈R s×(s−k) is orthogonal andR new =R C ∈ R(s−k)×(s−k) is the new upper triangular factor
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