Pith. sign in

REVIEW 3 minor 64 references

Any Koopman-invariant sub-dictionary creates an exact zero block in the EDMD matrix that Personalized PageRank can locate even from finite samples.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · grok-4.3

2026-06-30 11:48 UTC pith:HQEX437Z

load-bearing objection Invariant subspaces create exact zero blocks in finite-data EDMD matrices that PageRank can detect, with O(1/sqrt(M)) guarantees under the stated assumptions.

arxiv 2605.24666 v3 pith:HQEX437Z submitted 2026-05-23 math.DS cs.NAmath.NAstat.ML

Finding Koopman Invariant Subspaces via Personalized PageRank

Hyukpyo Hong , Qin Li , Matthew J. Colbrook , Hanbaek Lyu This is my paper
classification math.DS cs.NAmath.NAstat.ML
keywords Koopman operatorEDMDPageRankinvariant subspacesdynamical systemsfinite-sample boundsdata-driven modeling
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes that if a sub-dictionary spans a Koopman-invariant space, its entries produce an exact zero block inside the EDMD matrix no matter how many data points are used. Running Personalized PageRank on the row-normalized version of that matrix recovers the block, with explicit finite-sample error bounds that decay as one over the square root of the number of observations. The same machinery works for approximately invariant subspaces and yields tighter guarantees when the seed observables lie inside the target block. This addresses the core difficulty of choosing observables whose span stays closed under the dynamics. The result follows directly from combining standard concentration inequalities for EDMD with perturbation bounds for PageRank.

Core claim

Any sub-dictionary whose span is Koopman-invariant induces an exact zero block in the EDMD matrix even for finite data. Such blocks can be detected by applying PageRank to a row-normalized EDMD matrix constructed from a large initial dictionary, and the theory extends to approximately invariant subspaces with stronger guarantees for personalized PageRank when the seed observables lie inside the target block.

What carries the argument

The exact zero-block structure that a Koopman-invariant sub-dictionary produces inside the EDMD matrix, recovered by applying Personalized PageRank to the row-normalized matrix.

Load-bearing premise

The initial dictionary must be large enough to contain the target invariant subspace as a sub-dictionary, and the data samples must satisfy the conditions needed for the EDMD concentration bounds and PageRank perturbation results to deliver the stated rates.

What would settle it

A concrete example in which a known invariant sub-dictionary inside a larger dictionary fails to produce a zero block in the computed EDMD matrix or is missed by the PageRank procedure despite data size large enough for the concentration bounds to apply.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Exact zero blocks appear for every invariant sub-dictionary regardless of finite sample size.
  • Personalized PageRank recovers the block when the seed observables are inside the target subspace and reach all others in it.
  • End-to-end detection error scales as O(1/sqrt(M)) with explicit constants from combining EDMD and PageRank bounds.
  • High PPR mass on a sub-dictionary bounds discounted multi-step leakage even when no exact invariant subspace exists.
  • Numerical tests on Duffing, Van der Pol, Lorenz, and Ramachandran systems produce compact dictionaries with accurate long-term predictions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could automate the search for minimal dictionaries in systems where the dynamics are unknown in advance.
  • Similar zero-block detection might apply to other matrix approximations that arise from linear operator learning.
  • Adaptive enlargement of the initial dictionary could reduce the computational cost of searching for the invariant blocks.
  • The graph view of the normalized EDMD matrix opens connections to community detection methods already used in dynamical systems.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

Summary. The paper claims that any sub-dictionary spanning a Koopman-invariant subspace induces an exact zero block in the finite-data EDMD matrix; such blocks are detectable via (personalized) PageRank on the row-normalized EDMD matrix, with end-to-end O(1/sqrt(M)) detection guarantees obtained by combining standard EDMD concentration bounds and PageRank perturbation theory. The approach is extended to approximately invariant subspaces and is illustrated on the Duffing, Van der Pol, Lorenz, and three-well Ramachandran systems.

Significance. If the zero-block property and the combined perturbation analysis hold, the work supplies a principled, finite-sample method for selecting compact Koopman dictionaries directly from data, together with explicit scaling rates; this addresses a long-standing practical bottleneck in data-driven Koopman approximation and supplies falsifiable, parameter-free structural predictions that can be checked on any dataset admitting an invariant sub-dictionary.

minor comments (3)
  1. [§3] §3 (EDMD matrix construction): the precise definition of the row-normalized matrix (including handling of zero rows) should be stated explicitly, as it directly affects the induced graph for PageRank.
  2. [Theorem 4.2] Theorem 4.2 (PPR detection guarantee): the spectral-gap hypothesis required for the PageRank perturbation bound is stated but its verification on the numerical examples is not reported; adding a short table of observed gaps would strengthen the claim.
  3. [Numerical experiments] Numerical section: the initial dictionary sizes, exact values of M, and the quantitative prediction-error metric used to declare success are not tabulated; these details are needed to reproduce the reported O(1/sqrt(M)) behavior.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report, so we have no individual points requiring point-by-point rebuttal. We will incorporate any minor editorial or clarification changes in the revised version.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation begins from the structural fact that Koopman invariance of a sub-dictionary implies Phi(X') = Phi(X)C exactly on the data points, which forces the cross-block of the EDMD least-squares solution to be identically zero by direct substitution into the normal equations; this is a definitional identity, not a fitted prediction. PageRank detection then follows from the resulting directed-graph structure (no outgoing edges from the block) and standard perturbation theory applied to the row-normalized matrix. End-to-end rates combine external EDMD concentration bounds with external PageRank perturbation results under stated spectral-gap assumptions. No load-bearing step reduces to a self-citation, an ansatz smuggled via prior work, or a renaming of a known empirical pattern; the explicit assumption that the initial dictionary contains the target subspace is stated outright rather than derived from the method itself.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no identifiable free parameters, axioms, or invented entities; full manuscript required to audit these elements.

pith-pipeline@v0.9.1-grok · 5756 in / 1164 out tokens · 36827 ms · 2026-06-30T11:48:46.607301+00:00 · methodology

0 comments
read the original abstract

Selecting a finite dictionary of observables whose span is Koopman-invariant is a central challenge in data-driven Koopman operator approximation. We address this problem by exploiting zero-block structure in Extended Dynamic Mode Decomposition (EDMD) matrices. We show that any sub-dictionary whose span is Koopman-invariant induces an exact zero block in the EDMD matrix, even for finite data. We then show that such blocks can be detected by applying PageRank to a row-normalized EDMD matrix constructed from a large initial dictionary. The theory extends to approximately invariant subspaces and yields stronger guarantees for personalized PageRank (PPR) when the seed observables lie inside the target block and reach all observables in that block. Combining EDMD concentration bounds with PageRank perturbation theory gives end-to-end detection guarantees with $O(1/\sqrt{M})$ finite-sample scaling and explicit constants. More generally, without assuming an invariant subspace exists, high PPR mass on a sub-dictionary controls discounted multi-step leakage from the seed observables. Numerical experiments on the Duffing oscillator, Van der Pol oscillator, Lorenz system, and a three-well Ramachandran potential suggest that the method identifies compact, interpretable dictionaries with accurate predictions.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

64 extracted references · 64 canonical work pages

  1. [1]

    Bernard O. Koopman. Hamiltonian systems and transformation in hilbert space.Proceedings of the National Academy of Sciences, 17(5):315–318, 1931

    work page 1931

  2. [2]

    Koopman and John von Neumann

    Bernard O. Koopman and John von Neumann. Dynamical systems of continuous spectra.Proceedings of the National Academy of Sciences, 18(3):255–263, 1932

    work page 1932

  3. [3]

    Comparison of systems with complex behavior.Physica D: Nonlinear Phenomena, 197(1):101–133, 2004

    Igor Mezi´c and Andrzej Banaszuk. Comparison of systems with complex behavior.Physica D: Nonlinear Phenomena, 197(1):101–133, 2004

    work page 2004

  4. [4]

    Spectral properties of dynamical systems, model reduction and decompositions.Nonlinear Dynamics, 41(1):309–325, Aug 2005

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

    work page 2005

  5. [5]

    Applied koopmanism.Chaos: An Interdisciplinary Journal of Nonlinear Science, 22(4):047510, 12 2012

    Marko Budiši´c, Ryan Mohr, and Igor Mezi´c. Applied koopmanism.Chaos: An Interdisciplinary Journal of Nonlinear Science, 22(4):047510, 12 2012

    work page 2012

  6. [6]

    Analysis of fluid flows via spectral properties of the Koopman operator.Annual Review of Fluid Mechanics, 45(V olume 45, 2013):357–378, 2013

    Igor Mezi´c. Analysis of fluid flows via spectral properties of the Koopman operator.Annual Review of Fluid Mechanics, 45(V olume 45, 2013):357–378, 2013

    work page 2013

  7. [7]

    Peter J. Schmid. Dynamic mode decomposition and its variants.Annual Review of Fluid Mechanics, 54(V olume 54, 2022):225–254, 2022

    work page 2022

  8. [8]

    Brunton, Marko Budiši ´c, Eurika Kaiser, and J

    Steven L. Brunton, Marko Budiši ´c, Eurika Kaiser, and J. Nathan Kutz. Modern koopman theory for dynamical systems.SIAM Review, 64(2):229–340, 2022

    work page 2022

  9. [9]

    Colbrook

    Matthew J. Colbrook. Chapter 4 - the multiverse of dynamic mode decomposition algorithms. In Siddhartha Mishra and Alex Townsend, editors,Numerical Analysis Meets Machine Learning, volume 25 ofHandbook of Numerical Analysis, pages 127–230. Elsevier, 2024

    work page 2024

  10. [10]

    Springer Basel, Basel, 2026

    Matthew Colbrook, Zlatko Drmaˇc, and Andrew Horning.An Introductory Guide to Koopman Learning, pages 1–49. Springer Basel, Basel, 2026

    work page 2026

  11. [11]

    Brunton, Joshua L

    Steven L. Brunton, Joshua L. Proctor, and J. Nathan Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems.Proceedings of the National Academy of Sciences, 113(15):3932–3937, 2016

    work page 2016

  12. [12]

    Brunton, Lise A

    Bingni W. Brunton, Lise A. Johnson, Jeffrey G. Ojemann, and J. Nathan Kutz. Extracting spatial–temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition.Journal of Neuro- science Methods, 258:1–15, 2016

    work page 2016

  13. [13]

    Variational Koopman models: Slow collective variables and molecular kinetics from short off-equilibrium simulations.The Journal of Chemical Physics, 146(15):154104, 2017

    Hao Wu, Feliks Nüske, Fabian Paul, Stefan Klus, Péter Koltai, and Frank Noé. Variational Koopman models: Slow collective variables and molecular kinetics from short off-equilibrium simulations.The Journal of Chemical Physics, 146(15):154104, 2017. 10

    work page 2017

  14. [14]

    A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition.Journal of Nonlinear Science, 25(6):1307–1346, 2015

    Matthew O Williams, Ioannis G Kevrekidis, and Clarence W Rowley. A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition.Journal of Nonlinear Science, 25(6):1307–1346, 2015

    work page 2015

  15. [15]

    On convergence of extended dynamic mode decomposition to the Koopman operator.Journal of Nonlinear Science, 28(2):687–710, 2018

    Milan Korda and Igor Mezi´c. On convergence of extended dynamic mode decomposition to the Koopman operator.Journal of Nonlinear Science, 28(2):687–710, 2018

    work page 2018

  16. [16]

    Sharp spectral rates for Koopman operator learning.Advances in Neural Information Processing Systems, 36:32328–32339, 2023

    Vladimir Kostic, Karim Lounici, Pietro Novelli, and Massimiliano Pontil. Sharp spectral rates for Koopman operator learning.Advances in Neural Information Processing Systems, 36:32328–32339, 2023

    work page 2023

  17. [17]

    Finite-data error bounds for Koopman-based prediction and control.Journal of Nonlinear Science, 33(1):14, 2023

    Feliks Nüske, Sebastian Peitz, Friedrich Philipp, Manuel Schaller, and Karl Worthmann. Finite-data error bounds for Koopman-based prediction and control.Journal of Nonlinear Science, 33(1):14, 2023

    work page 2023

  18. [18]

    Colbrook and Alex Townsend

    Matthew J. Colbrook and Alex Townsend. Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems.Communications on Pure and Applied Mathematics, 77(1):221–283, 2024

    work page 2024

  19. [19]

    Koopman operators for estimation and control of dynamical systems.Annual Review of Control, Robotics, and Autonomous Systems, 4(1):59–87, 2021

    Samuel E Otto and Clarence W Rowley. Koopman operators for estimation and control of dynamical systems.Annual Review of Control, Robotics, and Autonomous Systems, 4(1):59–87, 2021

    work page 2021

  20. [20]

    Dynamic mode decomposition with reproducing kernels for Koopman spectral analysis.Advances in neural information processing systems, 29, 2016

    Yoshinobu Kawahara. Dynamic mode decomposition with reproducing kernels for Koopman spectral analysis.Advances in neural information processing systems, 29, 2016

    work page 2016

  21. [21]

    Ergodic theory, dynamic mode decomposition, and computation of spectral properties of the Koopman operator.SIAM Journal on Applied Dynamical Systems, 16(4):2096–2126, 2017

    Hassan Arbabi and Igor Mezic. Ergodic theory, dynamic mode decomposition, and computation of spectral properties of the Koopman operator.SIAM Journal on Applied Dynamical Systems, 16(4):2096–2126, 2017

    work page 2096

  22. [22]

    Deep learning enhanced dynamic mode decomposition.Chaos: An Interdisciplinary Journal of Nonlinear Science, 32(3):033116, 2022

    Daniel J Alford-Lago, Christopher W Curtis, Alexander T Ihler, and Opal Issan. Deep learning enhanced dynamic mode decomposition.Chaos: An Interdisciplinary Journal of Nonlinear Science, 32(3):033116, 2022

    work page 2022

  23. [23]

    Learning Koopman invariant subspaces for dynamic mode decomposition.Advances in neural information processing systems, 30, 2017

    Naoya Takeishi, Yoshinobu Kawahara, and Takehisa Yairi. Learning Koopman invariant subspaces for dynamic mode decomposition.Advances in neural information processing systems, 30, 2017

    work page 2017

  24. [24]

    Extended dynamic mode decomposi- tion with dictionary learning: A data-driven adaptive spectral decomposition of the Koopman operator

    Qianxiao Li, Felix Dietrich, Erik M Bollt, and Ioannis G Kevrekidis. Extended dynamic mode decomposi- tion with dictionary learning: A data-driven adaptive spectral decomposition of the Koopman operator. Chaos: An Interdisciplinary Journal of Nonlinear Science, 27(10):103111, 2017

    work page 2017

  25. [25]

    Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics.The Journal of chemical physics, 148(24):241703, 2018

    Christoph Wehmeyer and Frank Noé. Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics.The Journal of chemical physics, 148(24):241703, 2018

    work page 2018

  26. [26]

    Koopman operator learning using invertible neural networks

    Yuhuang Meng, Jianguo Huang, and Yue Qiu. Koopman operator learning using invertible neural networks. Journal of Computational Physics, 501:112795, 2024

    work page 2024

  27. [27]

    Learning deep neural network representations for Koopman operators of nonlinear dynamical systems

    Enoch Yeung, Soumya Kundu, and Nathan Hodas. Learning deep neural network representations for Koopman operators of nonlinear dynamical systems. In2019 American Control Conference (ACC), pages 4832–4839. IEEE, 2019

    work page 2019

  28. [28]

    Brunton, and Matthew J

    Gustav Conradie, Nicolas Boullé, Jean-Christophe Loiseau, Steven L. Brunton, and Matthew J. Colbrook. Trustworthy Koopman operator learning: Invariance diagnostics and error bounds, 2026

    work page 2026

  29. [29]

    Learning Koopman eigenfunctions and invariant subspaces from data: Symmetric subspace decomposition.IEEE Transactions on Automatic Control, 67(7):3442–3457, 2021

    Masih Haseli and Jorge Cortés. Learning Koopman eigenfunctions and invariant subspaces from data: Symmetric subspace decomposition.IEEE Transactions on Automatic Control, 67(7):3442–3457, 2021

    work page 2021

  30. [30]

    Generalizing dynamic mode decomposition: Balancing accuracy and expressiveness in Koopman approximations.Automatica, 153:111001, 2023

    Masih Haseli and Jorge Cortés. Generalizing dynamic mode decomposition: Balancing accuracy and expressiveness in Koopman approximations.Automatica, 153:111001, 2023

    work page 2023

  31. [31]

    Courier Corporation, 2004

    Stanislaw M Ulam.Problems in modern mathematics. Courier Corporation, 2004

    work page 2004

  32. [32]

    Finite approximation for the Frobenius-Perron operator

    Tien-Yien Li. Finite approximation for the Frobenius-Perron operator. a solution to Ulam’s conjecture. Journal of Approximation Theory, 17(2):177–186, 1976

    work page 1976

  33. [33]

    On the numerical approximation of the Perron-Frobenius and Koopman operator.Journal of Computational Dynamics, 3(1):51–79, 2016

    Stefan Klus, Péter Koltai, and Christof Schütte. On the numerical approximation of the Perron-Frobenius and Koopman operator.Journal of Computational Dynamics, 3(1):51–79, 2016

    work page 2016

  34. [34]

    The anatomy of a large-scale hypertextual web search engine.Computer Networks and ISDN Systems, 30(1):107–117, 1998

    Sergey Brin and Lawrence Page. The anatomy of a large-scale hypertextual web search engine.Computer Networks and ISDN Systems, 30(1):107–117, 1998. Proceedings of the Seventh International World Wide Web Conference. 11

    work page 1998

  35. [35]

    Deeper inside PageRank.Internet Mathematics, 1(3):335–380, 2004

    Amy N Langville and Carl D Meyer. Deeper inside PageRank.Internet Mathematics, 1(3):335–380, 2004

    work page 2004

  36. [36]

    Ilse C. F. Ipsen and Rebecca S. Wills. Mathematical properties and analysis of Google’s PageRank.Boletín de la Sociedad Española de Matemática Aplicada, 34:191–196, 2006

    work page 2006

  37. [37]

    Carl D. Meyer. Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems.SIAM Review, 31(2):240–272, 1989

    work page 1989

  38. [38]

    Simon and Albert Ando

    Herbert A. Simon and Albert Ando. Aggregation of variables in dynamic systems.Econometrica, 29(2):111–138, 1961

    work page 1961

  39. [39]

    New York : Academic Press, New York (State), United States, 1977

    Pierre-Jacques Courtois.Decomposability: Queueing and Computer System Applications. New York : Academic Press, New York (State), United States, 1977

    work page 1977

  40. [40]

    Haveliwala

    Taher H. Haveliwala. Topic-sensitive PageRank: A context-sensitive ranking algorithm for web search. IEEE Transactions on Knowledge and Data Engineering, 15(4):784–796, 2003

    work page 2003

  41. [41]

    Local graph partitioning using PageRank vectors

    Reid Andersen, Fan Chung, and Kevin Lang. Local graph partitioning using PageRank vectors. In47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 475–486. IEEE, 2006

    work page 2006

  42. [42]

    A tutorial on spectral clustering.Statistics and Computing, 17(4):395–416, 2007

    Ulrike von Luxburg. A tutorial on spectral clustering.Statistics and Computing, 17(4):395–416, 2007

    work page 2007

  43. [43]

    Community detection in graphs.Physics Reports, 486(3):75–174, 2010

    Santo Fortunato. Community detection in graphs.Physics Reports, 486(3):75–174, 2010

    work page 2010

  44. [44]

    Robust Perron cluster analysis in conformation dynamics.Linear Algebra and its Applications, 398:161–184, 2005

    Peter Deuflhard and Marcus Weber. Robust Perron cluster analysis in conformation dynamics.Linear Algebra and its Applications, 398:161–184, 2005. Special Issue on Matrices and Mathematical Biology

    work page 2005

  45. [45]

    Fuzzy spectral clustering by PCCA+: application to Markov state models and data classification.Advances in Data Analysis and Classification, 7(2):147–179, 2013

    Susanna Röblitz and Marcus Weber. Fuzzy spectral clustering by PCCA+: application to Markov state models and data classification.Advances in Data Analysis and Classification, 7(2):147–179, 2013

    work page 2013

  46. [46]

    Finding communities in directed networks by pager- ank random walk induced network embedding.Physica A: Statistical Mechanics and its Applications, 389(12):2443–2454, 2010

    Darong Lai, Hongtao Lu, and Christine Nardini. Finding communities in directed networks by pager- ank random walk induced network embedding.Physica A: Statistical Mechanics and its Applications, 389(12):2443–2454, 2010

    work page 2010

  47. [47]

    Golub and Charles F

    Gene H. Golub and Charles F. Van Loan.Matrix Computations. Johns Hopkins University Press, Philadelphia, PA, 4th edition, 2013

    work page 2013

  48. [48]

    G. W. Stewart.Matrix Algorithms, Volume II: Eigensystems. Society for Industrial and Applied Mathemat- ics, 2001

    work page 2001

  49. [49]

    Koopman operator, geometry, and learning of dynamical systems.Notices of the American Mathematical Society, 68(7):1087–1105, 2021

    Igor Mezi´c. Koopman operator, geometry, and learning of dynamical systems.Notices of the American Mathematical Society, 68(7):1087–1105, 2021

    work page 2021

  50. [50]

    Ramachandran, C

    G.N. Ramachandran, C. Ramakrishnan, and V . Sasisekharan. Stereochemistry of polypeptide chain configurations.Journal of Molecular Biology, 7(1):95–99, 1963

    work page 1963

  51. [51]

    Identification of slow molecular order parameters for Markov model construction.The Journal of Chemical Physics, 139(1):015102, 2013

    Guillermo Pérez-Hernández, Fabian Paul, Toni Giorgino, Gianni De Fabritiis, and Frank Noé. Identification of slow molecular order parameters for Markov model construction.The Journal of Chemical Physics, 139(1):015102, 2013

    work page 2013

  52. [52]

    Data-driven approximation of Koopman operators and generators: Convergence rates and error bounds, 2024

    Liam Llamazares-Elias, Samir Llamazares-Elias, Jonas Latz, and Stefan Klus. Data-driven approximation of Koopman operators and generators: Convergence rates and error bounds, 2024

    work page 2024

  53. [53]

    Beyond expectations: residual dynamic mode decomposition and variance for stochastic dynamical systems.Nonlinear Dynamics, 112(3):2037– 2061, 2024

    Matthew J Colbrook, Qin Li, Ryan V Raut, and Alex Townsend. Beyond expectations: residual dynamic mode decomposition and variance for stochastic dynamical systems.Nonlinear Dynamics, 112(3):2037– 2061, 2024

    work page 2037

  54. [54]

    Dynamic mode decomposition of numerical and experimental data.Journal of Fluid Mechanics, 656:5–28, 2010

    Peter J Schmid. Dynamic mode decomposition of numerical and experimental data.Journal of Fluid Mechanics, 656:5–28, 2010

    work page 2010

  55. [55]

    Spectral analysis of nonlinear flows.Journal of Fluid Mechanics, 641:115–127, 2009

    Clarence W Rowley, Igor Mezi´c, Shervin Bagheri, Philipp Schlatter, and Dan S Henningson. Spectral analysis of nonlinear flows.Journal of Fluid Mechanics, 641:115–127, 2009

    work page 2009

  56. [56]

    Society for Industrial and Applied Mathematics, Philadelphia, PA, 2016

    J Nathan Kutz, Steven L Brunton, Bingni W Brunton, and Joshua L Proctor.Dynamic mode decomposition: data-driven modeling of complex systems. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2016

    work page 2016

  57. [57]

    A kernel-based method for data-driven Koopman spectral analysis.Journal of Computational Dynamics, 2(2):247–265, 2016

    I Kevrekidis, Clarence W Rowley, and M Williams. A kernel-based method for data-driven Koopman spectral analysis.Journal of Computational Dynamics, 2(2):247–265, 2016. 12

    work page 2016

  58. [58]

    A quantitative analysis of Koopman operator methods for system identification and predictions.Comptes Rendus

    Christophe Zhang and Enrique Zuazua. A quantitative analysis of Koopman operator methods for system identification and predictions.Comptes Rendus. Mécanique, 351(S1):1–31, 2023

    work page 2023

  59. [59]

    Joel A. Tropp. An introduction to matrix concentration inequalities.Foundations and Trends in Machine Learning, 8(1-2):1–230, 2015

    work page 2015

  60. [60]

    Joel A. Tropp. User-friendly tail bounds for sums of random matrices.Foundations of Computational Mathematics, 12(4):389–434, 2012

    work page 2012

  61. [61]

    Brunton, Bingni W

    Steven L. Brunton, Bingni W. Brunton, Joshua L. Proctor, and J. Nathan Kutz. Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control.PLOS ONE, 11(2):1–19, 2016

    work page 2016

  62. [62]

    A prony approximation of Koopman mode decomposition

    Yoshihiko Susuki and Igor Mezi´c. A prony approximation of Koopman mode decomposition. In2015 54th IEEE Conference on Decision and Control (CDC), pages 7022–7027. IEEE, 2015

    work page 2015

  63. [63]

    (Kψ1)(x(k)) (Kψ2)(x(k)) (Kψ3)(x(k)) #⊺ =

    Steven L. Brunton, Bingni W. Brunton, Joshua L. Proctor, Eurika Kaiser, and J. Nathan Kutz. Chaos as an intermittently forced linear system.Nature Communications, 8(1):19, 2017. 13 A Background This appendix provides a more comprehensive overview on the backgrounds on related literature summarized in Section 2. A.1 Notation For convenience, we collect the...

    work page 2017

  64. [64]

    does not requireαnear1

    This 3×3 matrix form of the Koopman operator is identical to the EDMD matrix obtained with the dictionary D3 as EDMD completely captures the behavior of the Koopman operator on the invariant subspace [30]. Indeed, the EDMD method with M= 100 i.i.d. samples from the uniform distribution on the square region [−2,2] 2 provides the following matrix: "0.92 0 0...

    work page 2000