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REVIEW 2 major objections 5 minor 22 references

Koopman Subspace Pruning in Reproducing Kernel Hilbert Spaces via Principal Vectors

T0 review · 2 major / 5 minor · reviewed 2026-07-13 · grok-4.5

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Pith's one-line read Principal angles computed inside an RKHS let you prune Koopman dictionaries so the remaining subspace stays nearly invariant under the dynamics.

desk verdict Solid, usable RKHS extension of SPV pruning; the algebra is clean under a transparent idealization, and the Nyström construction is carefully done. read the letter →

arxiv 2604.01459 v2 pith:KA5E3AL7 submitted 2026-04-01 eess.SY cs.SYstat.ML

Dhruv Shah , Jorge Cortes This is my paper · ORCID
classification eess.SYcs.SYstat.ML
keywords KoopmanoperatorkernelEDMDprincipalanglessubspacepruningRKHSNyströmapproximationinvarianceproximity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Data-driven Koopman models work by projecting infinite-dimensional nonlinear dynamics onto a finite dictionary of observables. Accuracy collapses when that dictionary is not nearly invariant. This paper shows how to measure and improve that invariance when the dictionary lives in a reproducing kernel Hilbert space rather than ordinary Euclidean space. The key step is an exact formula (and a scalable Nyström version) for the principal angles and vectors between a candidate subspace and its image under the true Koopman operator, all measured with the RKHS inner product. Once those angles are known, the largest-angle principal vector can be discarded and the process repeated until the residual angle falls below a chosen tolerance. The resulting Kernel-SPV and Approximate Kernel-SPV procedures therefore turn an arbitrary kernel dictionary into a smaller, better-aligned one, improving eigenfunction quality and multi-step prediction on the damped Duffing oscillator.

What carries the argument

Theorem 4.5 (and its Nyström counterpart): the cosine matrix M = (R_V^dagger)^T M_cross R_KV^dagger formed from the QR factors of the Gram matrices; its singular values are exactly the cosines of the principal angles between S and KS inside the RKHS.

What would settle it

On a system and kernel where the residual of KV outside span(Phi_X) is deliberately large, check whether the principal angles returned by Algorithm 1 (or Algorithm 2) still match the angles obtained by an independent high-accuracy quadrature of the true RKHS inner products; systematic discrepancy would refute the claim that the computed angles equal the true invariance proximity.

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Extended reading notes

Core claim

Principal angles and vectors between a finite subspace S and its Koopman image KS can be computed exactly with respect to the RKHS inner product by constructing the three Gram matrices of the coefficient representations, extracting orthonormal QR factors via truncated eigendecompositions, and reading the cosines from the SVD of the resulting cosine matrix. The same geometry can be recovered at linear cost in the data size by replacing the full kernel matrix with a Nyström feature map and Tikhonov-regularized targets, yielding the Kernel-SPV and Approximate Kernel-SPV pruning algorithms.

Load-bearing premise

The finite data set must already be rich enough that every function obtained by applying the true dynamics to the dictionary still lies inside the span of the kernel sections centered at those same data points.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper develops a data-driven framework for computing principal angles and vectors between a finite-dimensional subspace S = span(V) and its Koopman image KS with respect to the RKHS inner product, thereby enabling subspace pruning (Kernel-SPV and Approximate Kernel-SPV) that reduces invariance proximity δ(S) = sin heta_max(S, KS). Under the idealized containment Assumption 3.3, Lemma 4.1 recovers the coefficient matrix W_KV, Gram matrices (12) are formed, QR factors are obtained via spectral truncation (Lemma 4.3), and principal angles/vectors are extracted by SVD of the cosine matrix (Theorem 4.5, Algorithm 1). A Nyström feature-map approximation with Tikhonov-regularized target matrices and rank-separation thresholds yields a scalable O(NDs + D^{3}) routine (Algorithm 2). Simulations on the damped Duffing oscillator show residual decay with landmark count D and improved multi-step eigenfunction prediction after pruning.

Significance. If the computational claims hold, the work usefully unifies the data-dependent dictionaries of Kernel EDMD with the geometric pruning already available in the Euclidean setting, providing a practical route to more invariant reduced models for prediction and control. The algebraic path from the definition of principal angles through the cosine-matrix SVD (Theorem 4.5) is clean under the stated assumption, the Nyström construction is carefully regularized, and the algorithms are fully specified. The contribution is incremental rather than foundational—the SPV loop itself is imported from prior Euclidean work—but the RKHS realization fills a genuine methodological gap and is of clear interest to the Koopman-control community.

major comments (2)
  1. Assumption 3.3 (Idealized Dataset Invariance) is load-bearing for the entire exact pipeline: Lemma 4.1, the construction of W_KV, the Gram matrices (12), and therefore Theorem 4.5 all require span(KV) ⊆ span(Φ_X). Remark 3.4 only argues that the residual can be driven small by enlarging N; it does not quantify the effect of a nonzero residual on the recovered principal angles. Without such a bound (or a numerical diagnostic that measures the residual of KV outside span(Φ_X)), it remains unclear how much of the reported “exact” angles are already approximate before Nyström is introduced. A short perturbation analysis or residual-norm experiment would strengthen the claim.
  2. Section V introduces free parameters (Tikhonov λ, rank-separation thresholds τ_V(D), τ_KV(D)) that are described as “tuning parameters set to constant multiples of D^{-1/2}” but are never given explicit values or selection rules in the Duffing experiments. Because these thresholds directly determine the ranks r_V and r_KV that enter the cosine matrix (23), their influence on the pruned subspace dimension s* and on the reported principal-angle match (Fig. 1, right) is material. Either fix them by a reproducible criterion or report a short sensitivity study.
minor comments (5)
  1. Abstract and Introduction speak of “randomized Nyström approximations,” yet Section V samples landmarks uniformly (or via clustering). Clarify whether randomness is essential or merely one possible selection strategy.
  2. Section VI title and text refer to the “damped Duffing oscillator,” while the Conclusions call it “undamped.” The discrete map (25) has no linear damping term; correct the terminology for consistency.
  3. Figure 1 (right) caption claims that Approximate Kernel-SPV with D = 2000 “closely matches the exact baseline,” but no quantitative distance (e.g., max | heta_i - hetã_i| or subspace angle) is reported. Adding a single scalar would make the visual claim more precise.
  4. The relation to the concurrent Principal Angle Decomposition (PAD) work [15] is noted only briefly in the Introduction. A short paragraph contrasting the two algorithms (exact vs. residual-based, pruning loop vs. one-shot ordering) would help readers place the contribution.
  5. Notation: the same symbol K is used for the infinite-dimensional Koopman operator and for kernel matrices; a typographic distinction (e.g., script K for the operator) would improve readability.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity; RKHS principal-angle formulas are standard linear algebra on kernel Gram matrices under an explicit idealization, with only benign self-citation of the authors' prior Euclidean SPV procedure.

  1. self citation load bearing [Section III (Problem 3.1) and Introduction]
    "The recently proposed Single Principal Vector (SPV) pruning procedure [10] provides a systematic solution to this search problem. This iterative method eliminates the principal vector associated with the largest principal angle at each step…"

    The pruning loop that the paper enables is taken wholesale from the authors' own concurrent Euclidean SPV work [10]. This is ordinary cumulative research rather than a circular reduction of the central claim (the RKHS computation of the principal vectors themselves), so it raises the score only to 1.

full rationale

The derivation chain (Assumption 3.3 o Lemma 4.1 for W_KV o Gram matrices (12) o spectral QR recovery Lemma 4.3 o cosine-matrix SVD Theorem 4.5) is algebraically self-contained and reduces to the classical Björck–Golub procedure once the RKHS inner products are realized via the kernel trick. The Nyström route (Algorithm 2) is an explicit low-rank approximation of the same objects, not a fit later re-labeled as a prediction. Simulation metrics (orthonormality residuals o D, multi-step eigenfunction error before/after pruning) are independent of the quantities used to select landmarks or regularization. Self-citations to the authors' Euclidean SPV papers ([9],[10]) and invariance-proximity definitions ([7],[8]) supply the pruning loop that is being enabled, but do not force the correctness of the new RKHS principal-vector formulas; those formulas stand on their own under the stated (idealized) containment. No self-definitional loop, no fitted-input-as-prediction, and no uniqueness theorem imported to forbid alternatives appear. Score 1 reflects only the minor, non-load-bearing self-citation of the SPV procedure itself.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard Hilbert-space geometry plus one idealized finite-data containment assumption and a handful of numerical tuning parameters required by the Nyström route. No new physical entities are postulated.

free parameters (5)
  • Nyström landmark count D
    Chosen by the user (tested values 800–4000); controls approximation quality and cost; not derived from first principles.
  • Tikhonov regularization λ
    Appears in the regularized least-squares solve for Z_KV; must be set by hand to stabilize the D×D inversion.
  • rank-separation thresholds τ_V(D), τ_KV(D)
    Hard cut-offs (suggested as multiples of D^{-1/2}) that decide which singular values of the target matrices are retained; directly affect the reported ranks and angles.
  • kernel smoothness β and support radius of Wendland kernel
    Hyper-parameters of the RKHS that define the geometry; fixed to β=2 in the experiments without systematic selection.
  • dictionary size s and coefficient matrix W_V
    Initial subspace dimension and its embedding into the full kernel sections; chosen by random sampling of centers.
assumptions (4)
  • standard math Principal angles between two subspaces of a Hilbert space are well-defined via the recursive max-inner-product characterization and can be recovered from the SVD of the cosine matrix of orthonormal bases (Björck–Golub).
    Invoked throughout Section II-B and Theorem 4.5.
  • standard math The RKHS associated with a continuous positive-definite kernel is a Hilbert space of functions with the reproducing property, so inner products of kernel sections reduce to kernel evaluations.
    Used to define all Gram matrices (12) and the feature map (19).
  • domain assumption Assumption 3.3: for the collected data, span(KV) ⊆ span(Φ_X).
    Stated explicitly; enables the linear solve for W_KV and all subsequent exact Gram matrices.
  • domain assumption The Nyström approximation K_{X,X} ≈ Ψ(X)^T Ψ(X) converges in the large-D limit for kernels with rapidly decaying spectra.
    Cited from the Nyström literature; underpins the claim that Approximate Kernel-SPV recovers the exact angles.

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Pith. "Pith review of Koopman Subspace Pruning in Reproducing Kernel Hilbert Spaces via Principal Vectors." pith.science (2026). https://pith.science/paper/KA5E3AL7

@misc{pith2026260401459,
  author       = {Pith},
  title        = {Pith review of: Koopman Subspace Pruning in Reproducing Kernel Hilbert Spaces via Principal Vectors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KA5E3AL7}},
  note         = {Machine review of arXiv:2604.01459}
}
read the original abstract

Data-driven approximations of the infinite-dimensional Koopman operator rely on finite-dimensional projections, where the predictive accuracy of the resulting models hinges heavily on the invariance of the chosen subspace. Subspace pruning systematically discards geometrically misaligned directions to enhance this invariance proximity, which formally corresponds to the largest principal angle between the subspace and its image under the operator. Yet, existing techniques are largely restricted to Euclidean settings. To bridge this gap, this paper presents an approach for computing principal angles and vectors to enable Koopman subspace pruning within a Reproducing Kernel Hilbert Space (RKHS) geometry. We first outline an exact computational routine, which is subsequently scaled for large datasets using randomized Nystrom approximations. Based on these foundations, we introduce the Kernel-SPV and Approximate Kernel-SPV algorithms for targeted subspace refinement via principal vectors. Simulation results validate our approach.

Figures

Figures reproduced from arXiv: 2604.01459 by the authors.

Figure 1
Figure 1. (Left) Residuals of the approximate orthonormal bases for [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Relative prediction error |K5ϕ−λ 5ϕ| for the estimated eigenfunction ϕ corresponding to λ ≈ 1, obtained from Kernel EDMD (left) and after pruning with Approximate Kernel-SPV (right). VII. CONCLUSIONS We have introduced a rigorous, data-driven framework for Koopman subspace pruning within an RKHS. We derived an exact computational routine to calculate principal angles and vectors. To resolve the computational bottlen… view at source ↗

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