REVIEW 3 major objections 5 minor 1 cited by
Koopman Operator for Stability Analysis: Theory with a Linear--Radial Product Reproducing Kernel
T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash
Pith's one-line read A learned Koopman operator can certify whether an equilibrium point is stable, using a linear–radial product kernel.
desk verdict The linear–radial product kernel is a genuinely new construction, but the stability certificate is unproven: Theorem 8 misses the spectrum at 0 for non-surjective f, and the error-bound argument uses an H→L2 norm where an H→H norm is required. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the linear–radial product kernel κ(x,x') = xᵀx' · ρ(|x−x'|), where xᵀx' is the linear kernel and ρ is the Wendland radial kernel. This product kernel defines an RKHS H whose members are finite sums of linear coordinate functions multiplied by Sobolev functions, making the space simultaneously regular enough for Koopman invariance and locally linear near the equilibrium. The spectral analysis then uses a homeomorphism to a linear system, reducing the Koopman spectrum to products of the Jacobian eigenvalues, and the stability certificate follows from a perturbation bound between the true and learned operators combined with spectral perturbation theory for self-adjoint ope
What would settle it
Find a smooth nonlinear system with an asymptotically stable equilibrium that satisfies the paper's smoothness and homeomorphism conditions, but where the learned Koopman operator (computed via kernel EDMD with the linear–radial product kernel, sufficient sample size, and appropriately chosen rank) has a largest eigenvalue magnitude ≥ 1−ε with high probability. Alternatively, exhibit a diagonalizable non-normal Koopman operator A (as in Lemma 12's setting) for which the derived bound on the spectral radius of A*A fails, invalidating the perturbation argument.
Extended reading notes
Core claim
The central claim is that the spectrum of the Koopman operator defined on the RKHS specified by the product kernel κ = κ⁻κ= (linear kernel times Wendland radial kernel) directly reflects the stability of the equilibrium at the origin. Specifically, assuming a Φ^s-homeomorphism conjugating the dynamics to a linear map, the Koopman spectrum equals the multiplicative semigroup generated by the eigenvalues of the Jacobian at the origin. Consequently, asymptotic stability forces the spectrum into the open unit disk, while instability places at least one eigenvalue outside. For the learned finite-rank operator  from kernel EDMD, Theorem 13 states that if |λ₁(Â)| + (ε_{n,δ} + 2||A||)ε_{n,δ} < 1, t
Load-bearing premise
The stability certificate relies on the assumption that the dynamics are conjugated to a linear system by a suitable homeomorphism, and on an unproven step where an H→L2 error bound is used as if it were an H→H error bound to control the spectral perturbation of  relative to A.
Editorial extensions
If this is right
- If the learned Koopman spectrum lies strictly inside the unit circle with the stated margin, one can certify asymptotic stability of the equilibrium without prior stability knowledge, purely from data.
- A spectrum escaping the unit circle can serve as a data-driven bifurcation detector, flagging when the equilibrium has become unstable.
- The spectrum–stability correspondence opens a route to Koopman-based control synthesis via kernel Lyapunov equations of the form V(f(x))−V(x) = −w(x), where V is a kernel quadratic form.
- The framework naturally extends to continuous-time systems through Koopman semigroups, potentially giving Lyapunov certificates for nonlinear control systems.
- The algebraic structure of the RKHS (locally linear, globally regular) may support learning Lyapunov functions that are locally quadratic but globally generic, bridging local stability analysis and global dynamics.
Reading between the lines
- The paper's homeomorphism assumption effectively restricts applicability to systems with a single global invariant structure; when an additional attractor like a limit cycle lies in the state space, the learned spectrum reflects the attractor rather than the equilibrium, as App. B shows. A natural extension would be to design kernels that encode multiple invariant structures so the spectrum disamb
- The stability certificate requires choosing the Wendland kernel smoothness parameter k in concert with the dynamics' smoothness; a testable practical extension is an adaptive scheme that selects k from data and checks whether the certificate remains valid, potentially using the error bound's constants.
- The proposed kernel Lyapunov equation (Eq. 12) is conjectural in the paper; a concrete next step would be to prove existence and uniqueness of its solution P and to bound the estimation error of the resulting Lyapunov function when  replaces A, which would turn the framework into a data-driven stability certificate with a computable Lyapunov function.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes a reproducing kernel Hilbert space (RKHS) construction—the product of a linear kernel and a Wendland radial kernel—on which the Koopman operator of a nonlinear discrete-time system is well-defined and bounded. The authors claim that, under a homeomorphic conjugacy to a linear system, the Koopman spectrum is the semigroup generated by the eigenvalues of the linearization, so that asymptotic stability of the equilibrium is characterized by the spectrum lying in the open unit disk. They then combine a kernel EDMD error bound with spectral perturbation estimates to argue that the finite-rank empirical Koopman operator can provide a probabilistic stability certificate. The theoretical claims are supported by proofs in an appendix and by numerical experiments on a Van der Pol oscillator.
Significance. If the central claims were fully established, the paper would contribute a concrete kernel design that makes the RKHS 'aware' of an equilibrium point, thereby connecting Koopman spectra to stability. The product-kernel idea is novel and potentially useful for Koopman-based control. The paper is honest about strong assumptions (global conjugacy to a linear map, finite hitting time, etc.) and includes a useful appendix discussion of why radial-only kernels cannot provide certificates. However, the main stability-certificate theorem rests on several unproven or incorrect technical steps. The paper does not provide code or machine-checked proofs, and the numerical experiments are illustrative rather than predictive, as they use data whose stability properties are already known. The result, as written, does not meet the standard for a rigorous stability certificate.
major comments (3)
- [Section 4.3, before Lemma 11] Lemma 10 bounds only the operator norm from H to L^2(X): ||Â−A||_{H→L²} ≤ ε_{n,δ}. The subsequent displayed inequality ||T−T̂|| ≤ (ε_{n,δ}+2||A||)ε_{n,δ} and the application of Lemma 11 require an H→H bound on ||Â−A||. On a compact domain the H norm dominates L², not conversely; an operator can be small in H→L² while having H→H norm of order 1 (e.g., a rank-one operator built from a highly oscillatory unit vector in H). The paper never establishes the needed H→H error bound, so the chain leading to Theorem 13 is broken.
- [Lemma 12 and Appendix A.6] The proof infers that ||Ah|| ≤ max_{λ∈σ(A)} |λ| · ||h|| from the fact that A is diagonal with respect to the monomial basis. This is only valid if the basis is orthonormal (or A is normal). The monomials are not orthogonal in the H inner product, and the proof gives no normality property. Consequently, the spectral radius of A*A is not shown to be |λ₁(F)|², and the discrete-spectrum/contractivity claim needed for Lemma 11 is unsupported. This is a load-bearing gap for the stability certificate.
- [Theorem 8 and Appendix A.4] The stated equality σ(A)=⟨σ(F)⟩ is not correct as written. When F is stable with |λ_i|<1, finite products λ_1^{α_1}...λ_d^{α_d} accumulate at 0, so 0 lies in the closure of this set and hence in the spectrum of A (as a limit point), even though 0∉⟨σ(F)⟩. The proof claims (λI−A)^{-1} exists for all λ∉⟨σ(F)⟩; for λ=0 this is false because the eigenvalues accumulate at 0. The correct statement would be σ(A)=closure(⟨σ(F)⟩), which still gives σ(A)⊂D for stable F, but the theorem as stated is false.
minor comments (5)
- [Abstract] The phrase 'ifwell-defined' should be 'if well-defined'.
- [Section 3.1] The notation 'κ =' and 'κ −' is visually confusing; consider renaming the kernels (e.g., κ_lin and κ_rad).
- [Section 5] The numerical section reports only qualitative trajectory and spectrum plots. Quantitative metrics (e.g., prediction error, convergence of the spectrum with n and r) would strengthen the verification. The statement that the spectrum 'overflows the unit circle' for μ=+1 is explicitly dependent on the sampling domain avoiding the limit cycle; this limitation is only discussed in Appendix B and should be flagged in the main text.
- [Section 6] The control-theoretic discussion is speculative and contains no theorems; it would be clearer to present it as future work rather than as a contribution.
- [Conclusion] The claim that 'such a certification did not exist with existing RKHS constructions' is too strong. Existing weighted-space or contractive Koopman constructions can provide stability certificates under different assumptions.
Circularity Check
No significant circularity: the product-kernel derivation and the stability certificate are not equivalent to their inputs; self-citations are contextual only.
full rationale
The paper's central claim is the construction of a product RKHS and the proof that, under explicit smoothness and homeomorphism assumptions, the Koopman spectrum is confined to the unit disk when the equilibrium is asymptotically stable (Theorem 8), with a finite-sample certificate (Theorem 13). This derivation is not circular: the kernel κ = κ−κ= is defined from the linear kernel and the Wendland radial kernel without fitting any parameter to the target stability conclusion; the spectrum–stability relation is obtained from an assumed conjugation to a linear map and polynomial density, not from the learned operator. The data-driven part relies on an external error bound (Lemma 10, from Philipp et al. 2024) and standard spectral perturbation theory (Lemma 11, from Lax 2002), so Theorem 13 is a logical consequence of the stated assumptions rather than a prediction forced by construction. The main weakness noted in the derivation is that Lemma 10 bounds Â−A only in the H→L2 norm while Lemma 11 is applied as if an H→H bound were available; this is a proof gap and a correctness risk, not circularity, because the conclusion does not reduce to the definition of any fitted quantity. The numerical examples use systems whose stability is already known and are therefore illustrative, not fitted-input predictions. Self-citations (Tang 2025a,b) are contextual references for weighted spaces and bilinear operator representations and are not load-bearing for Theorem 13. Explicit limitations are flagged in the text, e.g., Section 6: 'A rigorous theory on the solution existence, uniqueness, and learning error is left for future effort,' and Appendix B: 'the observations here cannot be theoretically guaranteed'; these are honest scope statements rather than disguised circular assumptions. No circular step satisfying the evidentiary standard of the review was found.
Assumptions & free parameters
free parameters (2)
- Wendland kernel order k =
not specified
- EDMD rank r and regularization β =
not reported
assumptions (6)
- domain assumption X is compact with Lipschitz boundary; f ∈ C^s and inf_x |det Df(x)| > 0.
- ad hoc to paper There exists a Φ^s-homeomorphism ψ conjugating f to a linear map z ↦ Fz on the whole state space.
- ad hoc to paper For the unstable case, sup_x τ(x) < ∞ and f^{-1} ∈ (Φ^s)^d.
- domain assumption Sample snapshots are i.i.d. uniformly distributed on X.
- ad hoc to paper The H→L2 error bound from Lemma 10 controls the H→H operator norm of A−Â.
- ad hoc to paper A is diagonal on a basis of monomials and satisfies ||Ah|| ≤ |λ1(F)| ||h||.
Cite this review
Pith. "Pith review of Koopman Operator for Stability Analysis: Theory with a Linear--Radial Product Reproducing Kernel." pith.science (2026). https://pith.science/paper/42XG6AIC
@misc{pith2026251106063,
author = {Pith},
title = {Pith review of: Koopman Operator for Stability Analysis: Theory with a Linear--Radial Product Reproducing Kernel},
year = {2026},
howpublished = {\url{https://pith.science/paper/42XG6AIC}},
note = {Machine review of arXiv:2511.06063}
}
read the original abstract
Koopman operator, as a fully linear representation of nonlinear dynamical systems, if well-defined on a reproducing kernel Hilbert space (RKHS), can be efficiently learned from data. For stability analysis and control-related problems, it is desired that the defining RKHS of the Koopman operator should account for both the stability of an equilibrium point (as a local property) and the regularity of the dynamics on the state space (as a global property). To this end, we show that by using the product kernel formed by the linear kernel and a Wendland radial kernel, the resulting RKHS is invariant under the action of Koopman operator (under certain smoothness conditions). Furthermore, when the equilibrium is asymptotically stable, the spectrum of Koopman operator is provably confined inside the unit circle, and escapes therefrom upon bifurcation. Thus, the learned Koopman operator with provable probabilistic error bound provides a stability certificate. In addition to numerical verification, we further discuss how such a fundamental spectrum--stability relation would be useful for Koopman-based control.
Figures
Figures from the paper (2 more)
Forward citations
Cited by 1 Pith paper
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Data-Driven Domain of Attraction Estimation via Convergent Koopman-Zubov Approximation
A new RKHS construction gives the Zubov-Koopman operator a spectrum inside the unit disk, so the Zubov function can be estimated from data with a sectorially bounded error.
Reference graph
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Reviewed August 3, 2026 · model on record in the stance chip above.
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