Pith. sign in

REVIEW 3 major objections 4 minor 4 cited by

Convergent Methods for Koopman Operators on Reproducing Kernel Hilbert Spaces

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper introduces the first provably convergent, data-driven algorithms for Koopman spectral analysis on reproducing kernel Hilbert spaces, and proves no algorithm can use fewer limits.

desk verdict Strong new convergence and optimality results for RKHS Koopman methods, with a real but narrow scope gap for unbounded operators. read the letter →

arxiv 2506.15782 v1 pith:6UTF3JMQ submitted 2025-06-18 math.NA cs.LGcs.NAmath.DSmath.SPstat.ML

Nicolas Boullé , Matthew J. Colbrook , Gustav Conradie This is my paper · ORCID
classification math.NAcs.LGcs.NAmath.DSmath.SPstat.ML MSC 37A3037M1037N1047A1047B3247B3365P99
keywords KoopmanoperatorPerron–FrobeniusreproducingkernelHilbertspacedata-drivenspectralanalysispseudospectraverifiedeigenpairsanderrorcontrolsolvabilitycomplexityindexmeasures
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

This paper claims to settle what can and cannot be computed when the Koopman operator of a dynamical system is studied on a reproducing kernel Hilbert space (RKHS). It introduces the first general family of data-driven algorithms for computing spectra, pseudospectra, and spectral measures of Koopman and Perron-Frobenius operators on such spaces, each with convergence proofs and, for eigenpairs and predictions, computable error bounds. The technical reason the methods work is that on an RKHS the adjoint operator applied to a kernel function is again a kernel function, $K^*K_x = K_{F(x)}$, so all inner products needed for a Galerkin approximation are exact kernel evaluations; this removes the large-data limit that $L^2$-based methods provably require. Against the algorithms stand matching impossibility results in the Solvability Complexity Index hierarchy, showing that no algorithm, deterministic or randomized, can succeed with fewer successive limits or with full error control. If the claims hold, kEDMD-style computations can be certified and are provably optimal in their limit structure, and the choice of kernel becomes the main modelling decision.

What carries the argument

The load-bearing object is the reproducing kernel $K_x$ together with the identity $K^*K_x = K_{F(x)}$: applying the Perron-Frobenius adjoint to a kernel function just moves the kernel to the pushed-forward state. From snapshot pairs $(x_k, F(x_k))$, three matrices are formed, namely $G_{jk}=K(x_k,x_j)$, $A_{jk}=K(F(x_k),x_j)$, and $R_{jk}=K(F(x_k),F(x_j))$, so that the finite-section approximation of $K^*$ is $G^{-1}A$ and the squared residual of a candidate pair $(\lambda,g)$ is the Rayleigh quotient $g^*(R-\lambda A^* - \bar\lambda A + |\lambda|^2 G)g / (g^*G g)$, both exact and limit-free. The convergence argument runs through the injection modulus: the smallest generalized eigenvalue of these matrices decreases to $\sigma_{\inf}(K^*-zI)$ as $N\to\infty$, giving single-limit convergence for the adjoint's approximate point pseudospectrum, while the Koopman operator itself needs a second limit via rectangular truncations. The same constructions supply the adversarial interval-exchange and weighted-sequence dynamics used in the impossibility proofs.

What would settle it

The test is to run the residual-based pseudospectrum algorithm on an unbounded Koopman operator for which kernel functions provably fail to form a core of $K^*$, for instance a composition operator on a Sobolev space $H^r((0,1))$ whose domain includes functions that no finite kernel span can approximate in the graph norm. If the computed residual stays bounded away from $\sigma_{\inf}(K^*-zI)$ at a genuine spectral point $z$ for every dictionary size $N$, the single-limit convergence guarantee breaks exactly where the core assumption fails; checking the same question on the paper's own unbounded adversarial maps from Theorem 4.18 would show whether the upper and lower bounds cover the same class of systems.

Watch

Extended reading notes

Core claim

The paper's central claim is that working in a reproducing kernel Hilbert space makes the spectral analysis of Koopman operators both certifiable and cheaper than in the classical $L^2$ setting. On an RKHS, the Perron-Frobenius adjoint $K^*$ acts on kernel functions by the exact formula $K^*K_x = K_{F(x)}$, so the Galerkin matrices $G$, $A$, and $R$ built from kernel evaluations are available with no large-data quadrature limit. From these matrices the authors construct residual-based algorithms, namely SpecRKHS-Eig for verified eigenpairs, SpecRKHS-PseudoPF and SpecRKHS-PseudoKoop for approximate point pseudospectra of $K^*$ and $K$, SpecRKHS-Obs for error-controlled observable prediction, and spectral-measure algorithms for unitary and self-adjoint cases, and prove convergence as the number of kernel dictionary points $N$ grows. They then prove optimality in the Solvability Complexity Index hierarchy: $\mathrm{Sp}_{\mathrm{ap},\epsilon}(K^*)\in\Sigma^G_1\setminus\Delta^G_1$, $\mathrm{Sp}_{\mathrm{ap}}(K^*)\in\Pi^G_2\setminus\Delta^G_2$, $\mathrm{Sp}_{\mathrm{ap},\epsilon}(K)\in\Sigma^G_2\setminus\Delta^G_2$, and $\mathrm{Sp}_{\mathrm{ap}}(K)\in\Pi^G_3\setminus\Delta^G_3$, meaning no algorithm, including randomized ones, can use fewer successive limits or obtain full error control. The method is demonstrated on turbulent channel flow, molecular dynamics of a binding protein, Antarctic sea ice concentration, and Northern Hemisphere sea surface height.

Load-bearing premise

Everything rests on the assumption, introduced in Section 3.4 before Lemma 3.2 and used to prove Theorem 3.5, that a countable set of kernel functions spans a core of the Perron-Frobenius operator, meaning every function in its domain can be approximated together with its image by kernel combinations; this holds automatically for bounded operators, but for unbounded densely defined Koopman operators it is a real domain condition, and if it fails the kernel-based residual checks can miss spectrum.

Editorial extensions

If this is right

  • kEDMD output becomes checkable: any candidate eigenpair can be accepted or rejected by an exact RKHS residual, so spurious eigenvalues are filtered by a computable threshold rather than trusted from the discretization.
  • Because $G$, $A$, and $R$ are exact kernel evaluations, finite externally provided datasets are used directly, with no quadrature rule, ergodicity assumption, or large-data limit $M\to\infty$ required.
  • The limit counts are optimal: $\mathrm{Sp}_{\mathrm{ap},\epsilon}(K^*)$ needs one limit with one-sided verification, $\mathrm{Sp}_{\mathrm{ap}}(K^*)$ needs two, and the Koopman-side analogues need two and three; the paper constructs adversarial dynamics showing no deterministic or randomized algorithm can do better, with randomized success capped at probability $2/3$.
  • Observable prediction carries explicit error bounds through Perron-Frobenius mode decompositions, with a concrete duality rule: use Perron-Frobenius methods for many observables at few starting points and Koopman methods for few observables at many points.
  • For unitary or self-adjoint Koopman operators, characterized by kernel invariance conditions, scalar spectral measures are computable at high order with resolvent kernels, exposing continuous spectra that eigenvalue-based methods cannot see; the paper demonstrates this on Moebius rotations, random walks, turbulent channel flow, and molecular dynamics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the results hold, the kernel is not just an approximation device but a complexity dial: two different RKHSs can assign different Solvability Complexity Indices to the same dynamical system, so choosing a kernel is implicitly choosing how many limits an algorithm must take. A testable extension would be to map the SCI of $\mathrm{Sp}_{\mathrm{ap}}(K^*)$ across a family of kernels (Gaussian, Mate
  • The verification framework suggests a kernel-selection criterion the paper uses only informally: tune kernel parameters to minimize the verified residual of the modes of interest rather than only the forecast error; the adaptive scale search in the Duffing example is a prototype.
  • The impossibility results are built on Sobolev-type RKHSs (native spaces of Wendland and Matern kernels); if the same adversarial embeddings work in Gaussian or analytic kernel spaces, then even the smoothest popular kernels would not lower the SCI, extending the optimality claims to the most widely used settings.
  • The equivalence between deterministic and randomized impossibility transfers the certified-residual idea into a safety net for stochastic spectral methods: randomized dictionary learning cannot be expected to converge generically, but filtering its output through the paper's residuals would give it a verifiable certificate where it does succeed.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper develops data-driven spectral algorithms for Koopman and Perron–Frobenius operators on reproducing kernel Hilbert spaces (RKHSs). Its core observation is Eq. (3.1): the adjoint acts on kernel functions by K^*K_x = K_{F(x)}, which lets all Galerkin matrices and residuals be evaluated exactly from kernel evaluations, bypassing the quadrature/data limit M→∞ needed in L^2-based EDMD. On this basis, the paper proposes verified eigenpair computation (Algorithm 2), approximate point pseudospectrum computation for the Perron–Frobenius operator (Algorithm 3), a two-limit pseudospectrum algorithm for the Koopman operator (Algorithm 4), error-controlled observable prediction (Algorithm 5), and algorithms for spectral measures of self-adjoint and unitary operators (Algorithms 6 and 7). Convergence theorems are proved for noise-free and noisy data. The paper then formulates matching lower bounds in the Solvability Complexity Index hierarchy, constructing adversarial interval exchange maps on Sobolev spaces and discrete constructions on h^r(N), and proves that randomized algorithms do not bypass these lower bounds. The methods are demonstrated on high-dimensional examples including turbulent channel flow, molecular dynamics, Antarctic sea ice concentration, and sea surface height, with code made publicly available.

Significance. If the main claims hold, this is a substantial advance: it provides a certified, RKHS-native replacement for kEDMD-type spectral analysis, with residuals computed exactly from the kernel, convergence in a single limit for the Perron–Frobenius approximate point pseudospectrum, and SCI-optimality results including a randomized-algorithm extension. The clean derivation of K^*K_x = K_{F(x)}, the exact residual formulas, the careful treatment of noisy snapshots via kernel moduli of continuity, and the demonstration on genuinely high-dimensional examples are all strong positive features. The main caveat is that the central convergence theorems for unbounded operators depend on an unverified core assumption; until this assumption is either proved for relevant unbounded classes or explicitly incorporated into the claims, the paper's advertised scope as 'first general ... on RKHSs' is broader than what is actually established.

major comments (3)
  1. [Section 3.4, assumption before Lemma 3.2; Theorems 3.5 and 3.9] The convergence of Algorithm 3 to Sp_{ap,ϵ}(K^*) rests on the assumption that span{K_{x_i}: i∈N} is a core of K^*. This is load-bearing: Lemma 3.4's proof needs, for each g∈D(K^*), a sequence g_N∈V_N with g_N→g and K^*g_N→K^*g. Lemma 3.2 establishes the required density only for bounded K^*. For unbounded densely defined Koopman operators, the paper gives no criterion, no example, and no verification in the unbounded cases it later constructs. If the core assumption fails, the finite-dimensional residuals in Eq. (3.5) can remain bounded below by a positive gap to σ_inf(K^*-zI), so spectral invisibility may persist despite all residual checks passing. The same issue appears in Section 3.5.2, where 'we assume that {K_1,K_2,...} is a countable subset ... forms a core of K and K^*'. This must be fixed either by proving the core property for the unbounded classes considered or by explicitly restricting the convergence statements and the abstract's 'general' claim.
  2. [Section 4.4.1, proof of Theorem 4.18] The impossibility proof constructs a limiting interval exchange map F and states that 'it is, however, densely defined by considering compactly supported smooth functions, which are dense in H^r((0,1))'. Density of smooth functions makes K densely defined, but the proof does not show that the constructed F belongs to the input class Ω_{H^r((0,1))} defined in Section 4.2, which requires that the span of countably many kernel functions be a core of both K and K^*. For r>1 the constructed K_F is unbounded, so Lemma 3.2 does not apply. Without a core verification for the constructed example, the lower bound is stated for a class that is larger than the class for which the upper bound is proved, and the apparent matching of upper and lower bounds is incomplete.
  3. [Abstract and Section 1.4] The abstract's phrase 'the first general, provably convergent, data-driven algorithms for computing spectral properties of Koopman and Perron–Frobenius operators on RKHSs' and Section 1.4's statement 'These results apply to any RKHS' overstate the proven scope. The convergence theorems require the densely defined-operator setup of Section 2.4, the core assumption of Section 3.4, and, for the noisy-data extensions, a kernel modulus of continuity satisfying the conditions of Theorem 3.13. Large parts of the theory, including the spectral-measure algorithms in Section 5.3, are justified under additional boundedness or resolvent-convergence hypotheses. The claims should be reworded to state the precise conditions under which convergence is proved, or the missing conditions should be proved.
minor comments (4)
  1. [Section 2.5, Eq. (2.6)] The Attouch–Wets metric formula uses 2^{-n} but B_m(0) in the same term; the index should be consistent, i.e., B_n(0).
  2. [Theorem 3.13] The statement contains the typo 'snapshot date' where 'snapshot data' is meant; this appears in both the theorem statement and its proof.
  3. [Section 4.1.3, Definition 4.9] The definition of ∆_1-information writes 'for all ζ∈Ω' inside the description of Ω_∆1, but the intended meaning is that the approximations converge to f_j(ζ) for each ζ; the phrasing should be clarified.
  4. [Section 5.3.3] The statement that convergence theorems from [32,34,38] 'extend immediately' to Algorithms 6 and 7 is too quick for the self-adjoint case, where the finite-section method for the resolvent is noted not to converge in general; the text should specify precisely which convergence result Algorithm 6 carries.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction found; central algorithms are self-contained, with an explicit core assumption that limits the claimed generality on unbounded operators.

full rationale

Section 3.1 derives the exact identity K*K_x = K_F(x) from the reproducing property, and the Gram matrices G, A, R in Eqs. (3.2)-(3.4) are kernel evaluations; no parameter is fitted to a subset of data and then renamed as a prediction. The residual in Eq. (3.5) is a true finite-dimensional Rayleigh quotient, and Theorem 3.5 proves Attouch-Wets convergence of Gamma_epsilon_N(K*) to Sp_{ap,epsilon}(K*) via Lemma 3.4, which is an analytical monotone-convergence result rather than a definitional identity. The upper bounds in Theorem 4.10 are proven from these lemmas, and the lower bounds in Section 4 are independent adversarial constructions; the SCI optimality is not imported from the authors' prior work as a black box. The explicit assumption in Section 3.4 that span{K_x} is a core of K* for unbounded operators is a genuine restriction on the abstract's 'any RKHS' formulation, but it is an assumption, not a redefinition of the spectrum or a hidden fit. Lemma 4.32 is cited from [36], which overlaps with the present authors, and is used in Theorem 4.31, but it is a standard consistency property and the theorem's proof is otherwise self-contained, so this is a minor self-citation that is not load-bearing. The central derivation therefore has independent mathematical content and is not circular.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the standard RKHS framework plus the core assumption for unbounded operators. No new physical entities are introduced; the only hand-tuned quantity is the kernel scale in numerical examples.

free parameters (1)
  • Kernel scale sigma = e.g., sigma=6 for Duffing, sigma=1/2000 for MD, sigma=1/10000 for sea ice
    Chosen by grid search in numerical examples to improve matrix conditioning; not part of the convergence theorems and not fitted to the spectral sets.
assumptions (5)
  • standard math Moore-Aronszajn theorem: a positive definite kernel defines a unique RKHS
    Used in Section 2.3 to connect kernels and RKHSs.
  • standard math Sobolev embedding theorem for r > d/2 gives RKHS status of H^r
    Used in Section 2.4 to define the RKHS of Sobolev spaces and the cone condition.
  • domain assumption The span of kernel functions at countably many points forms a core of K*
    Assumed in Section 3.4 (before Lemma 3.2 and used in Theorem 3.5) to ensure residual computations reflect the full operator; automatic for bounded operators.
  • domain assumption Snapshot data satisfy y(m)=F(x(m)) exactly in the main convergence results
    Used in Section 3.4; relaxed in Section 3.7 under a kernel modulus of continuity.
  • domain assumption Spectral measure algorithms inherit convergence results from [32,34,38]
    Section 5.3.3 relies on prior work for resolvent convergence and does not reprove these rates.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Convergent Methods for Koopman Operators on Reproducing Kernel Hilbert Spaces." pith.science (2026). https://pith.science/paper/6UTF3JMQ

@misc{pith2026250615782,
  author       = {Pith},
  title        = {Pith review of: Convergent Methods for Koopman Operators on Reproducing Kernel Hilbert Spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6UTF3JMQ}},
  note         = {Machine review of arXiv:2506.15782}
}
abstract

Data-driven spectral analysis of Koopman operators is a powerful tool for understanding numerous real-world dynamical systems, from neuronal activity to variations in sea surface temperature. The Koopman operator acts on a function space and is most commonly studied on the space of square-integrable functions. However, defining it on a suitable reproducing kernel Hilbert space (RKHS) offers numerous practical advantages, including pointwise predictions with error bounds, improved spectral properties that facilitate computations, and more efficient algorithms, particularly in high dimensions. We introduce the first general, provably convergent, data-driven algorithms for computing spectral properties of Koopman and Perron--Frobenius operators on RKHSs. These methods efficiently compute spectra and pseudospectra with error control and spectral measures while exploiting the RKHS structure to avoid the large-data limits required in the $L^2$ settings. The function space is determined by a user-specified kernel, eliminating the need for quadrature-based sampling as in $L^2$ and enabling greater flexibility with finite, externally provided datasets. Using the Solvability Complexity Index hierarchy, we construct adversarial dynamical systems for these problems to show that no algorithm can succeed in fewer limits, thereby proving the optimality of our algorithms. Notably, this impossibility extends to randomized algorithms and datasets. We demonstrate the effectiveness of our algorithms on challenging, high-dimensional datasets arising from real-world measurements and high-fidelity numerical simulations, including turbulent channel flow, molecular dynamics of a binding protein, Antarctic sea ice concentration, and Northern Hemisphere sea surface height. The algorithms are publicly available in the software package $\texttt{SpecRKHS}$.

Figures

Figures reproduced from arXiv: 2506.15782 by the authors.

Figure 1
Figure 1. Schematic of the main contributions summarized in Tables 1 and 2. We compute approximations of Koop [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. To compute σinf(K − zI), we begin with a N1 × N2 truncation of K − zI from which we may compute σinf(PN1 (K−zI)P ∗ N2 ), take the limit as N1 → ∞ to compute σinf((K−zI)P ∗ N2 ), and then take the limit as N2 → ∞ to get σinf(K − zI). These two limits do not commute and generally cannot be combined into a single limit (e.g., by adaptively choosing N2 dependent on N1). 3.5.1 Quadrature If the samples are chosen to form… view at source ↗
Figure 3
Figure 3. Gauss map. Left: Relative forecast errors for the Gauss map for kEDMD compared to EDMD. Right: The [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figures from the paper (21 more)
Figure 4
Figure 4. Figure 4: Duffing oscillator. Comparison of relative errors for predicting trajectories of the Duffing oscillator for six [PITH_FULL_IMAGE:figures/full_fig_p024_4.png]
Figure 5
Figure 5. Figure 5: As for the Gauss map, there is spectral pollution in the eigenvalues outputted by kEDMD, with only those near [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]
Figure 5
Figure 5. Figure 5: Duffing oscillator. Left: The eigenvalues outputted by kEDMD, with the color showing the size of their [PITH_FULL_IMAGE:figures/full_fig_p025_5.png]
Figure 6
Figure 6. Figure 6: Lorenz system. Left: The eigenvalues outputted by kEDMD, with the color showing the size of their [PITH_FULL_IMAGE:figures/full_fig_p025_6.png]
Figure 7
Figure 7. Figure 7: Lorenz system. Six pseudoeigenfunctions for the chaotic Lorenz system with eigenvalue [PITH_FULL_IMAGE:figures/full_fig_p026_7.png]
Figure 8
Figure 8. Figure 8: Mobius map. Left: Approximation to spectral measure for ¨ T1; the rotation on the unit disk by π/3, plotted on a logarithmic scale. Right: Approximation to the spectral measure for the Mobius map defined by ¨ T2(z) = (az + b)/(bz + a) with a = √ 2e iπ√ 3 and b = e iπ9/…
Figure 9
Figure 9. Figure 9: Mobius map. Analysis of the rate of convergence of SpecRKHS-UniMeasure (Algorithm 7) to the true ¨ spectral measure for the Mobius map ¨ T2 at a single point z = π/3. N is fixed and we vary ϵ and m. The black dotted line show agreement with the theoretical convergence …
Figure 10
Figure 10. Figure 10: Random walk. Left: Approximation to spectral measure of system defined by Eq. (5.6) using SpecRKHS [PITH_FULL_IMAGE:figures/full_fig_p053_10.png]
Figure 11
Figure 11. Figure 11: Turbulent channel flow. Left: The eigenvalues outputted by kEDMD using a Mat [PITH_FULL_IMAGE:figures/full_fig_p054_11.png]
Figure 12
Figure 12. Figure 12: Turbulent channel flow. (a)-(c) Perron–Frobenius modes corresponding to three verified eigenfunctions for [PITH_FULL_IMAGE:figures/full_fig_p055_12.png]
Figure 13
Figure 13. Figure 13: Molecular dynamics (MD) dataset. Left: The eigenvalues outputted by kEDMD using a Mat [PITH_FULL_IMAGE:figures/full_fig_p056_13.png]
Figure 14
Figure 14. Figure 14: Molecular dynamics (MD) dataset. Approximation to the spectral measure of molecular dynamics system [PITH_FULL_IMAGE:figures/full_fig_p056_14.png]
Figure 15
Figure 15. Figure 15: Antarctic sea ice data. Left: The eigenvalues outputted by kEDMD using a Mat [PITH_FULL_IMAGE:figures/full_fig_p057_15.png]
Figure 16
Figure 16. Figure 16: Antarctic sea ice data. Perron–Frobenius modes for Antarctic sea ice data, along with their residuals [PITH_FULL_IMAGE:figures/full_fig_p057_16.png]
Figure 17
Figure 17. Figure 17: Antarctic sea ice data. Relative forecast errors for DMD, kEDMD, and SpecRKHS-Obs (Algorithm 5), [PITH_FULL_IMAGE:figures/full_fig_p058_17.png]
Figure 18
Figure 18. Figure 18: Antarctic sea ice data. Plots of exact and predicted Antarctic sea ice concentrations over a three-month [PITH_FULL_IMAGE:figures/full_fig_p058_18.png]
Figure 19
Figure 19. Figure 19: Northern Hemisphere sea surface height. Left: The eigenvalues outputted by kEDMD using a Mat [PITH_FULL_IMAGE:figures/full_fig_p060_19.png]
Figure 20
Figure 20. Figure 20: Northern Hemisphere sea surface height. Top: The Perron–Frobenius eigenvalue and mode corresponding [PITH_FULL_IMAGE:figures/full_fig_p060_20.png]
Figure 21
Figure 21. Figure 21: Relative forecast errors for DMD, kEDMD, and SpecRKHS-Obs (Algorithm 5), compared to the exact [PITH_FULL_IMAGE:figures/full_fig_p061_21.png]
Figure 22
Figure 22. Figure 22: Left: A comparison of predictions between exact simulation data, kEDMD, and SpecRKHS-Obs (Algo [PITH_FULL_IMAGE:figures/full_fig_p061_22.png]
Figure 23
Figure 23. Figure 23: Left: An examination of the sources of error when predicting future trajectories using SpecRKHS-Obs [PITH_FULL_IMAGE:figures/full_fig_p061_23.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A pseudospectral approach to rigorous numerical estimation of resonances of transfer operators

    math.DS 2025-07 conditional novelty 7.0 of 10

    A pseudospectral, computer-assisted framework yields certified enclosures and exclosures of Ruelle-Pollicott resonances for analytic expanding circle maps, demonstrated on a Blaschke benchmark and a perturbed doubling map.

  2. Koopman Subspace Pruning in Reproducing Kernel Hilbert Spaces via Principal Vectors

    eess.SY 2026-04 conditional novelty 6.0 of 10

    Kernel-SPV and Approximate Kernel-SPV compute RKHS principal angles/vectors to prune Koopman subspaces for better invariance proximity, with Nyström scaling for large data.

  3. Endpoint Koopman Spectral Computation: $L^1$ Residual Bounds, $L^\infty$ Instability, and Point-Spectral SCI Calibration Families

    math.LO 2026-01 conditional novelty 6.0 of 10

    For L∞ Koopman operators on Cantor systems, approximate point spectra are not computable by any finite tower of algorithms; L1 upper bounds match the reflexive regime.

  4. Avoiding spectral pollution for transfer operators using residuals

    math.DS 2025-07 conditional novelty 6.0 of 10

    A residual computation for kernelized dynamic mode decomposition gives a necessary condition for eigenvalues of transfer operators, enabling detection of spectral pollution.

Reference graph

Works this paper leans on

112 extracted references · 71 canonical work pages · cited by 4 Pith papers

  1. [1]

    R. A. Adams and J. J. F. Fournier.Sobolev Spaces. Elsevier, 2003

    2003

  2. [2]

    Aiena.Fredholm and local spectral theory, with applications to multipliers

    P. Aiena.Fredholm and local spectral theory, with applications to multipliers. Springer Science & Business Media, 2004

    2004

  3. [3]

    Aloisio, S

    M. Aloisio, S. L. Carvalho, C. R. de Oliveira, and E. Souza. On spectral measures and convergence rates in von Neumann’s Ergodic Theorem.Monatsh. Math., 203(3):543–562, 2024

    2024

  4. [4]

    J. W. Anderson.Hyperbolic Geometry. Springer Science & Business Media, 2006. 61

    2006

  5. [5]

    Aronszajn

    N. Aronszajn. Theory of reproducing kernels.Trans. Amer. Math. Soc., 68(3):337–404, 1950

    1950

  6. [6]

    O. F. Bandtlow, W. Just, and J. Slipantschuk. EDMD for expanding circle maps and their complex perturbations.Appl. Comput. Harmon. Anal., 73:101690, 2024

    2024

  7. [7]

    The extended Smale's 9th problem -- On computational barriers and paradoxes in estimation, regularisation, computer-assisted proofs and learning

    A. Bastounis, A. C. Hansen, and V . Vlaˇci´c. The extended Smale’s 9th problem–On computational barriers and paradoxes in estimation, regularisation, computer-assisted proofs and learning.arXiv preprint arXiv:2110.15734, 2021

    work page Pith review arXiv

  8. [8]

    Beckstein

    O. Beckstein. Molecular dynamics trajectory of I-FABP for testing and benchmarking solvent dynamics analy- sis.https://figshare.com/articles/dataset/Molecular_dynamics_trajectory_of_I-FABP_ for_testing_and_benchmarking_solvent_dynamics_analysis/7058030, 2018

    work page arXiv 2018

Show all 112 references
  1. [9]

    Beer.Topologies on closed and closed convex sets

    G. Beer.Topologies on closed and closed convex sets. Springer Science & Business Media, 1993

    1993

  2. [10]

    Ben-Artzi, M

    J. Ben-Artzi, M. J. Colbrook, A. C. Hansen, O. Nevanlinna, and M. Seidel. Computing spectra - On the solvability com- plexity index hierarchy and towers of algorithms.arXiv preprint arXiv:1508.03280, 2020

    2020 arXiv

  3. [11]

    Berlinet and C

    A. Berlinet and C. Thomas-Agnan.Reproducing kernel Hilbert spaces in probability and statistics. Springer Science & Business Media, 2011

    2011

  4. [12]

    Bevanda, M

    P. Bevanda, M. Beier, A. Lederer, S. Sosnowski, E. H ¨ullermeier, and S. Hirche. Koopman kernel regression. InAdvances in Neural Information Processing Systems, volume 36, pages 16207–16221, 2023

    2023

  5. [13]

    G. D. Birkhoff. Proof of the ergodic theorem.Proc. Natl. Acad. Sci. USA, 17(12):656–660, 1931

    1931

  6. [14]

    C. M. Bishop.Pattern recognition and machine learning. Springer, 2006

    2006

  7. [15]

    L. Blum, F. Cucker, M. Shub, and S. Smale.Complexity and Real Computation. Springer New York, 1998

    1998

  8. [16]

    L. Bold, F. M. Philipp, M. Schaller, and K. Worthmann. Kernel-based Koopman approximants for control: Flexible sam- pling, error analysis, and stability.arXiv preprint arXiv:2412.02811, 2024

    2024

  9. [17]

    B ¨ottcher and S

    A. B ¨ottcher and S. M. Grudsky.Toeplitz matrices, asymptotic linear algebra and functional analysis. Springer, 2000

    2000

  10. [18]

    B. W. Brunton, L. A. Johnson, J. G. Ojemann, and J. N. Kutz. Extracting spatial-temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition.J. Neurosci. Methods, 258:1–15, 2016

    2016

  11. [19]

    S. L. Brunton, M. Budi ˇsi´c, E. Kaiser, and J. N. Kutz. Modern Koopman theory for dynamical systems.SIAM Rev., 64(2):229–340, 2022

    2022

  12. [20]

    S. L. Brunton and J. N. Kutz.Data-driven Science and Engineering: Machine learning, Dynamical systems, and Control. Cambridge University Press, 2019

    2019

  13. [21]

    Budi ˇsi´c, R

    M. Budi ˇsi´c, R. Mohr, and I. Mezi´c. Applied Koopmanism.Chaos, 22(4):047510, 2012

    2012

  14. [22]

    B. J. Carswell, B. D. MacCluer, and A. Schuster. Composition operators on the Fock space.Acta Sci. Math., 69(3-4):871– 887, 2003

    2003

  15. [23]

    Chac ´on and J

    G. Chac ´on and J. Gim ´enez. Composition operators on spaces of entire functions.Proc. Amer. Math. Soc., 135(7):2205– 2218, 2007

    2007

  16. [24]

    Christmann and I

    A. Christmann and I. Steinwart.Support vector machines. Springer, 2008

    2008

  17. [25]

    M. J. Colbrook.The Foundations of Infinite-dimensional Spectral Computations. PhD thesis, University of Cambridge, 2020

    2020

  18. [26]

    M. J. Colbrook. Computing spectral measures and spectral types.Commun. Math. Phys., 384(1):433–501, 2021

    2021

  19. [27]

    M. J. Colbrook. On the computation of geometric features of spectra of linear operators on Hilbert spaces.Found. Comput. Math., pages 1–82, 2022

    2022

  20. [28]

    M. J. Colbrook. The mpEDMD algorithm for data-driven computations of measure-preserving dynamical systems.SIAM J. Numer. Anal., 61(3):1585–1608, 2023

    2023

  21. [29]

    M. J. Colbrook. Another look at Residual Dynamic Mode Decomposition in the regime of fewer Snapshots than Dictionary Size.Phys. D: Nonlinear Phenom., 469:134341, 2024

    2024

  22. [30]

    M. J. Colbrook.The multiverse of dynamic mode decomposition algorithms, volume 25 ofHandbook of Numerical Analysis, pages 127–230. Elsevier, 2024

    2024

  23. [31]

    M. J. Colbrook, L. J. Ayton, and M. Sz ˝oke. Residual dynamic mode decomposition: Robust and verified Koopmanism.J. Fluid Mech., 955:A21, 2023

    2023

  24. [32]

    M. J. Colbrook, C. Drysdale, and A. Horning. Rigged dynamic mode decomposition: Data-driven generalized eigenfunction decompositions for Koopman operators.SIAM J. Appl. Dyn. Syst., 24(2):1150–1190, 2025

    2025

  25. [33]

    M. J. Colbrook and A. C. Hansen. The foundations of spectral computations via the solvability complexity index hierarchy. J. Eur. Math. Soc., 25(12):4639–4728, 2022

    2022

  26. [34]

    M. J. Colbrook, A. Horning, and A. Townsend. Computing spectral measures of self-adjoint operators.SIAM Rev., 63(3):489–524, 2021

    2021

  27. [35]

    M. J. Colbrook, Q. Li, R. V . Raut, and A. Townsend. Beyond expectations: residual dynamic mode decomposition and variance for stochastic dynamical systems.Nonlinear Dyn., 112(3):2037–2061, 2024. 62

    2024

  28. [36]

    M. J. Colbrook, I. Mezi ´c, and A. Stepanenko. Limits and powers of Koopman learning.arXiv preprint arXiv:2407.06312, 2024

    2024 arXiv

  29. [37]

    M. J. Colbrook, B. Roman, and A. C. Hansen. How to compute spectra with error control.Phys. Rev. Lett., 122(25):250201, 2019

    2019

  30. [38]

    M. J. Colbrook and A. Townsend. Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems.Commun. Pure Appl. Math., 77(1):221–283, 2023

    2023

  31. [39]

    J. B. Conway.A course in functional analysis. Springer Science & Business Media, 1994

    1994

  32. [40]

    M. C. Cross and P. C. Hohenberg. Pattern formation outside of equilibrium.Rev. Mod. Phys., 65(3):851, 1993

    1993

  33. [41]

    S. Das, D. Giannakis, and J. Slawinska. Reproducing kernel Hilbert space compactification of unitary evolution groups. Appl. Comput. Harmon. Anal., 54:75–136, 2021

    2021

  34. [42]

    M. V . de Hoop, N. B. Kovachki, N. H. Nelsen, and A. M. Stuart. Convergence rates for learning linear operators from noisy data.SIAM-ASA J. Uncertain. Quantif., 11(2):480–513, 2023

    2023

  35. [43]

    Dellnitz and O

    M. Dellnitz and O. Junge. On the approximation of complicated dynamical behavior.SIAM J. Numer. Anal., 36(2):491–515, 1999

    1999

  36. [44]

    F. F. Di Bruno. Note sur une nouvelle formule de calcul diff ´erentiel.Quarterly J. Pure Appl. Math, 1:359–360, 1857

  37. [45]

    Di Nezza, G

    E. Di Nezza, G. Palatucci, and E. Valdinoci. Hitchhiker’s guide to the fractional Sobolev spaces.Bull. Sci. Math., 136(5):521–573, 2012

    2012

  38. [46]

    Eisner, B

    T. Eisner, B. Farkas, M. Haase, and R. Nagel.Operator Theoretic Aspects of Ergodic Theory. Springer International Publishing, 2015

    2015

  39. [47]

    L. C. Evans.Partial Differential Equations. American Mathematical Society, 2nd edition, 2010

    2010

  40. [48]

    Feller.An introduction to probability theory and its applications

    W. Feller.An introduction to probability theory and its applications. John Wiley & Sons, 1991

    1991

  41. [49]

    Fetterer, K

    F. Fetterer, K. Knowles, W. Meier, M. Savoie, and A. Windnagel. Sea Ice Index, Version 3, 2017.https://nsidc. org/data/G02135/versions/3

    2017

  42. [50]

    Froyland

    G. Froyland. On Ulam approximation of the isolated spectrum and eigenfunctions of hyperbolic maps.Discrete Contin. Dyn. Syst., 17(3):671–689, 2007

    2007

  43. [51]

    Froyland, D

    G. Froyland, D. Giannakis, B. R. Lintner, M. Pike, and J. Slawinska. Spectral analysis of climate dynamics with operator- theoretic approaches.Nature Commun., 12(1), 2021

    2021

  44. [52]

    Froyland, D

    G. Froyland, D. Giannakis, E. Luna, and J. Slawinska. Revealing trends and persistent cycles of non-autonomous systems with autonomous operator-theoretic techniques.Nature Commun., 15(1):4268, 2024

    2024

  45. [53]

    Giannakis

    D. Giannakis. Data-driven spectral decomposition and forecasting of ergodic dynamical systems.Appl. Comput. Harmon. Anal., 47(2):338–396, 2019

    2019

  46. [54]

    Giannakis and C

    D. Giannakis and C. Valva. Consistent spectral approximation of Koopman operators using resolvent compactification. Nonlinearity, 37(7):075021, 2024

    2024

  47. [55]

    Gonzalez, M

    E. Gonzalez, M. Abudia, M. Jury, R. Kamalapurkar, and J. A. Rosenfeld. The kernel perspective on dynamic mode decom- position.arXiv preprint arXiv:2106.00106, 2021

    2021 arXiv

  48. [56]

    Graham, K

    J. Graham, K. Kanov, X. Yang, M. Lee, N. Malaya, C. Lalescu, R. Burns, G. Eyink, A. Szalay, R. Moser, et al. A web services accessible database of turbulent channel flow and its use for testing a new integral wall model for LES.J. Turbul., 17(2):181–215, 2016

    2016

  49. [57]

    D. A. Haggerty, M. J. Banks, E. Kamenar, A. B. Cao, P. C. Curtis, I. Mezi ´c, and E. W. Hawkes. Control of soft robots with inertial dynamics.Sci. Robot., 8(81), 2023

    2023

  50. [58]

    Hales, M

    T. Hales, M. Adams, G. Bauer, T. D. Dang, J. Harrison, L. T. Hoang, C. Kaliszyk, V . Magron, S. McLaughlin, T. T. Nguyen, et al. A formal proof of the Kepler conjecture.Forum Math. Pi, 5:e2, 2017

    2017

  51. [59]

    P. R. Halmos.Lectures on ergodic theory. Courier Dover Publications, 2017

    2017

  52. [60]

    A. C. Hansen. On the solvability complexity index, then-pseudospectrum and approximations of spectra of operators.J. Am. Math. Soc., 24(1):81–124, 2011

    2011

  53. [61]

    Haroske and H

    D. Haroske and H. Triebel.Distributions, Sobolev Spaces, Elliptic Equations. European Mathematical Society, 2007

    2007

  54. [62]

    Hartman.Ordinary Differential Equations

    P. Hartman.Ordinary Differential Equations. SIAM, 2002

    2002

  55. [63]

    Hastie, R

    T. Hastie, R. Tibshirani, and J. Friedman.The Elements of Statistical Learning. Springer New York, 2009

    2009

  56. [64]

    T. Hey, S. Tansley, and K. M. Tolle.The Fourth Paradigm: Data-Intensive Scientific Discovery. Microsoft Research, 2009

    2009

  57. [65]

    J. Hogg, M. Fonoberova, and I. Mezi ´c. Exponentially decaying modes and long-term prediction of sea ice concentration using Koopman mode decomposition.Sci. Rep., 10(1):16313, 2020

    2020

  58. [66]

    R. A. Horn and C. R. Johnson.Matrix Analysis. Cambridge University Press, 2012

    2012

  59. [67]

    Ikeda, I

    M. Ikeda, I. Ishikawa, and C. Schlosser. Koopman and Perron–Frobenius operators on reproducing kernel Banach spaces. Chaos, 32(12), 2022. 63

    2022

  60. [68]

    JAMSTEC OFES (Ocean General Circulation Model for the Earth Simulator) Dataset, 2009.https://doi

    JAMSTEC. JAMSTEC OFES (Ocean General Circulation Model for the Earth Simulator) Dataset, 2009.https://doi. org/10.17596/0002029

    2009 doi

  61. [69]

    R. E. Kalman. Mathematical description of linear dynamical systems.J. Soc. Ind. Appl. Math., 1(2):152–192, 1963

    1963

  62. [70]

    Kato.Perturbation Theory for Linear Operators

    T. Kato.Perturbation Theory for Linear Operators. Springer, 1995

    1995

  63. [71]

    Kawahara

    Y . Kawahara. Dynamic mode decomposition with reproducing kernels for Koopman spectral analysis. InAdvances in Neural Information Processing Systems, volume 29, 2016

    2016

  64. [72]

    Klus and N

    S. Klus and N. D. Conrad. Dynamical systems and complex networks: A Koopman operator perspective.J. Phys. Complex., 5(4):041001, 2024

    2024

  65. [73]

    S. Klus, I. Schuster, and K. Muandet. Eigendecompositions of transfer operators in reproducing kernel Hilbert spaces.J. Nonlinear Sci., 30:283–315, 2020

    2020

  66. [74]

    K ¨ohne, F

    F. K ¨ohne, F. M. Philipp, M. Schaller, A. Schiela, and K. Worthmann.L∞-error Bounds for Approximations of the Koopman Operator by Kernel Extended Dynamic Mode Decomposition.SIAM J. Appl. Dyn. Syst., 24(1):501–529, 2025

    2025

  67. [75]

    B. O. Koopman. Hamiltonian systems and transformation in Hilbert space.Proc. Natl. Acad. Sci. USA, 17(5):315–318, 1931

    1931

  68. [76]

    B. O. Koopman and J. von Neumann. Dynamical systems of continuous spectra.Proc. Natl. Acad. Sci. USA, 18(3):255–263, 1932

    1932

  69. [77]

    Korda and I

    M. Korda and I. Mezi ´c. On convergence of extended dynamic mode decomposition to the Koopman operator.J. Nonlinear Sci., 28(2):687–710, 2018

    2018

  70. [78]

    Korda, M

    M. Korda, M. Putinar, and I. Mezi ´c. Data-driven spectral analysis of the Koopman operator.Appl. Comput. Harmon. Anal., 48(2):599–629, 2020

    2020

  71. [79]

    Kostic, K

    V . Kostic, K. Lounici, P. Novelli, and M. Pontil. Sharp spectral rates for Koopman operator learning. InAdvances in Neural Information Processing Systems, volume 36, pages 32328–32339, 2023

    2023

  72. [80]

    Kostic, P

    V . Kostic, P. Novelli, A. Maurer, C. Ciliberto, L. Rosasco, and M. Pontil. Learning dynamical systems via Koopman operator regression in reproducing kernel Hilbert spaces. InAdvances in Neural Information Processing Systems, volume 35, pages 4017–4031, 2022

    2022

  73. [81]

    Lewin and ´E

    M. Lewin and ´E. S´er´e. Spectral pollution and how to avoid it.Proc. London Math. Soc., 100(3):864–900, 2010

    2010

  74. [82]

    Lusch, J

    B. Lusch, J. N. Kutz, and S. L. Brunton. Deep learning for universal linear embeddings of nonlinear dynamics.Nature Commun., 9(1):1–10, 2018

    2018

  75. [83]

    Mauroy and I

    A. Mauroy and I. Mezic. Analytic Extended Dynamic Mode Decomposition.arXiv preprint arXiv:2405.15945, 2024

    2024

  76. [84]

    Mauroy, I

    A. Mauroy, I. Mezi ´c, and J. Moehlis. Isostables, isochrons, and Koopman spectrum for the action–angle representation of stable fixed point dynamics.Phys. D: Nonlinear Phenom., 261:19–30, 2013

    2013

  77. [85]

    N. G. Meyers and J. Serrin.H=W.Proc. Natl. Acad. Sci. USA, 51(6):1055–1056, 1964

    1964

  78. [86]

    I. Mezi ´c. Spectral properties of dynamical systems, model reduction and decompositions.Nonlinear Dyn., 41(1):309–325, 2005

    2005

  79. [87]

    I. Mezi ´c. Analysis of fluid flows via spectral properties of the Koopman operator.Annu. Rev. Fluid Mech., 45(1):357–378, 2013

    2013

  80. [88]

    I. Mezi ´c. Spectrum of the Koopman operator, spectral expansions in functional spaces, and state-space geometry.J. Nonlinear Sci., 30(5):2091–2145, 2020

    2020

  81. [89]

    Mohri, A

    M. Mohri, A. Rostamizadeh, and A. Talwalkar.Foundations of Machine Learning. MIT Press, 2nd edition, 2018

    2018

  82. [90]

    Nakatsukasa

    Y . Nakatsukasa. Absolute and relative Weyl theorems for generalized eigenvalue problems.Linear Algebra Appl., 432(1):242–248, 2010

    2010

  83. [91]

    J. v. Neumann. Proof of the quasi-ergodic hypothesis.Proc. Natl. Acad. Sci. USA, 18(1):70–82, 1932

    1932

  84. [92]

    N ¨uske, S

    F. N ¨uske, S. Peitz, F. Philipp, M. Schaller, and K. Worthmann. Finite-data error bounds for Koopman-based prediction and control.J. Nonlinear Sci., 33(1):14, 2023

    2023

  85. [93]

    M. O. Williams, C. W. Rowley, and I. G. Kevrekidis. A kernel-based method for data-driven Koopman spectral analysis.J. Comput. Dyn., 2(2):247–265, 2015

    2015

  86. [94]

    Orvieto et al

    A. Orvieto et al. Resurrecting recurrent neural networks for long sequences. InInternational Conference on Machine Learning, pages 26670–26698. PMLR, 2023

    2023

  87. [95]

    V . I. Paulsen and M. Raghupathi.An introduction to the theory of reproducing kernel Hilbert spaces. Cambridge University Press, 2016

    2016

  88. [96]

    Philipp, M

    F. Philipp, M. Schaller, K. Worthmann, S. Peitz, and F. N¨uske. Error analysis of kernel EDMD for prediction and control in the Koopman framework.arXiv preprint arXiv:2312.10460, 2023

    2023 arXiv

  89. [97]

    F. M. Philipp, M. Schaller, K. Worthmann, S. Peitz, and F. N ¨uske. Error bounds for kernel-based approximations of the Koopman operator.Appl. Comput. Harmon. Anal., 71:101657, 2024. 64

    2024

  90. [98]

    J. L. Proctor and P. A. Eckhoff. Discovering dynamic patterns from infectious disease data using dynamic mode decompo- sition.Int. Health, 7(2):139–145, 2015

    2015

  91. [99]

    Reed and B

    M. Reed and B. Simon.Methods of Modern Mathematical Physics, Volume I: Functional Analysis. Academic Press, 2nd edition, 1980

    1980

  92. [100]

    P. J. Schmid. Dynamic mode decomposition of numerical and experimental data.J. Fluid Mech., 656:5–28, 2010

    2010

  93. [101]

    Schmidt and H

    M. Schmidt and H. Lipson. Distilling free-form natural laws from experimental data.Science, 324(5923):81–85, 2009

    2009

  94. [102]

    Sch ¨olkopf

    B. Sch ¨olkopf. The kernel trick for distances. InAdvances in Neural Information Processing Systems, volume 13, pages 283–289, 2000

    2000

  95. [103]

    Schwinger

    J. Schwinger. Unitary operator bases.Proc. Natl. Acad. Sci. U.S.A., 46(4):570–579, 1960

    1960

  96. [104]

    Shargorodsky

    E. Shargorodsky. On the level sets of the resolvent norm of a linear operator.Bull. Lond. Math. Soc., 40(3):493–504, 2008

    2008

  97. [105]

    Skogestad and I

    S. Skogestad and I. Postlethwaite.Multivariable feedback control: analysis and design. John Wiley & Sons, 2005

    2005

  98. [106]

    S. H. Strogatz.Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. CRC Press, 2nd edition, 2018

    2018

  99. [107]

    Stuart and A

    A. Stuart and A. R. Humphries.Dynamical Systems and Numerical Analysis. Cambridge University Press, 1998

    1998

  100. [108]

    G. Szeg ˝o. Beitr ¨age zur Theorie der Toeplitzschen Formen.Math. Z., 6(3-4):167–202, 1920

    1920

  101. [109]

    L. N. Trefethen and M. Embree.Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Prince- ton University Press, 2005

    2005

  102. [110]

    Tucker.Validated Numerics

    W. Tucker.Validated Numerics. Princeton University Press, 2011

    2011

  103. [111]

    Wendland.Scattered data approximation

    H. Wendland.Scattered data approximation. Cambridge University Press, 2004

    2004

  104. [112]

    M. O. Williams, I. G. Kevrekidis, and C. W. Rowley. A data–driven approximation of the Koopman operator: Extending dynamic mode decomposition.J. Nonlinear Sci., 25(6):1307–1346, 2015. 65

    2015

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.