Koopman Kernel Regression

Summary

This paper proposes a Koopman eigenfunction kernel that induces a Koopman-invariant RKHS and the corresponding analytical solution of the regression problem with this kernel. They further provide theoretical guarantees and numerical experiments to demonstrate the efficacy of the proposed method.

Strengths

[1] Designing the kernel that induces a Koopman-invariant RKHS is a challenging problem, and this paper contributes to this problem. [2] The theoretical results in Section 4 guarantee the performance of the proposed method.

Weaknesses

[1] The presentation is poor, for example, there should be more clearly mathematical description about the Koopman approximation theory in RKHS and the claimed open challenge in the Koopman operator community; in Line 13, a noun is missing after 'aforementioned'. [2] The authors only test the proposed method in low-dimensional toy models.

Questions

[1] In Lemma 2, how to guarantee the Koopman eigenfunctions span the continuous functions space $C(\mathbb{X})$, or you cannot obtain the universality of eigenfunctions? [2] The way of omitting spatial modes in Eq. (5) is not standard, and even it can simplify the formulation in your mathematical deduction, the spatial modes are still required in numerical experiments to approximate any measurement function $q$. How do you represent any measurement $y_T=q(x_T)$ without coefficients in Eq. (5)? [3] In Lemma 1, the authors define $g$ as $g:\mathbb{X}_0\to C(\mathbb{X}_0)$, in this way $g$ is a functional not a function. However, $\phi_\lambda$ is the Koopman eigenfunction related to the original dynamics $f$ and $g$ is any functional, how do you guarantee the Eq. (4) stand? If the Lemma 1 was wrong, the Koopman-invariant property in Theorem 1 (iii) cannot stand either. [4] Why not compare your method with the kernel based Koopman operator learning methods such as [R1-3]? The EDMD and PCR are classic methods but their performance is now lagging behind. __References__ [R1] Baddoo, P. J., Herrmann, B., McKeon, B. J., & Brunton, S. L. (2022). Kernel learning for robust dynamic mode decomposition: linear and nonlinear disambiguation optimization. Proceedings of the Royal Society A, 478(2260), 20210830. [R2] Kostic, V., Novelli, P., Maurer, A., Ciliberto, C., Rosasco, L., & Pontil, M. (2022). Learning dynamical systems via koopman operator regression in reproducing kernel hilbert spaces. Advances in Neural Information Processing Systems, 35, 4017-4031. [R3] Meanti, G., Chatalic, A., Kostic, V. R., Novelli, P., Pontil, M., & Rosasco, L. (2023). Estimating Koopman operators with sketching to provably learn large scale dynamical systems. arXiv preprint arXiv:2306.04520.

Rating

4: Borderline reject: Technically solid paper where reasons to reject, e.g., limited evaluation, outweigh reasons to accept, e.g., good evaluation. Please use sparingly.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

2 fair

Presentation

2 fair

Contribution

2 fair

Limitations

[1] In reality, the Koopman eigenvalues and eigenfunctions are unknown and are exactly the learning objectives. However, this paper requires the Koopman eigenvalues to construct the Koopman kernel, which is impractical in real-world scenarios. [2] The authors only test their method in toy models instead of real-world dataset such as PIV data for flow over a cylinder or molecular dynamics.

Summary

This papers introduces a, to my knoweldge, new kernel on trajectories for dynamical system learning derived from Koopman operator theory. Consistency guarantees and generalisation bounds are derived using classical RKHS arguments. Numerical experiments on Bi-stable system and Van der Pol oscillator are performed using the proposed technique.

Strengths

The idea of making links between Koopman operator theory and kernel methods is interesting. The proposed kernel is to my knowledge new.

Weaknesses

There is a general lack of mathematical rigour throughout the paper (see my remarks and questions in the next section). Some statements are unjustified and at times untrue. For example in line 49-50 it is claimed that the RKHS associated to the derived kernel is the first in the literature that posseding universal features that have linear dynamics. *Signature kernels* [1] have such property and are universal for the much broader class of continuous functions on (rough) paths. Furthermore, the numerical examples are very simplistic and the experimental details are poorely described. The paper has good potential. I am willing to increase my rating if the authors are able to address my points below reagarding the mathematical rigour and provide a fair theoretical and numerical comparison with [1]. **References** [1] Salvi, Cristopher, et al. "The Signature Kernel is the solution of a Goursat PDE." SIAM Journal on Mathematics of Data Science 3.3 (2021): 873-899.

Questions

- For readers unfamiliar with the theory of dynamical systems it is worth reminding the definition of forward-complete system in line 74, perhaps as a footnote. - Equation (1b) is slightly ambiguous as the function $q$ acts pointwise on the path $x: t \mapsto x(t)$; it would be clearer to write $y(t) = q(x(t))$ for any $t \in \mathbb{T}$. - It is worth specifying what are the minimal regularity conditions required from $f$ and $q$ in (1a) and (1b) so that the flow in line 76 exists and is unique. - The map $g$ in equation (2a) is vaguely introduced as a "feature map"; can the authors provide more details regarding this map? - I think combining equations (3a) and (3b) into a single equation would make the optimisation objective clearer. - In lines 87-88 it is claimed that (3) *is generally not analytically solvable or tractable*; this is not true in general, as (3) is nothing but an ODE and numerous methods for fitting parameters of ODEs are available, including the recent Neural ODE framework [1] and many others. - In lines 137-138 it is mentioned that "the Koopman eigen-functions in Definition 1 are universal approximators of continuous functions"; can the authors give more details about this statement? What's the hypothesis space of continuous functions mentioned in this sentence? - Lemma 1 is vague and not rigorously stated: what is the space $\mathbb X_0$? Is $C(\mathbb X_0)$ the space of continuous functions from $\mathbb X_0$ to $\mathbb R$? What's the regularity of $g$? To talk about continuity one needs a topology on input and output spaces; what are the choice of topologies in this Lemma? What does the wording "... are assigned..." in line 146 mean precisely? The function $M$ implicitly depends on $c,g,A$ so I don't understand the claim of the Lemma. The norm $|| \cdot ||_{\mathbb Y_T}$ has not been been defined. - Point (ii) in Theorem 1: what do you mean by universal here? - It seems to me that the kernel defined in equation (10) does not take its values on the space $\mathcal{B}(\mathbb Y_T)$ of bounded linear operators on $\mathbb Y_T$. - Is the kernel defined in Proposition 1 positive (semi)definite? - Theorem 2 needs to be written in much higher level of details, recalling the definitions of $M, \mathbf \Gamma, \hat \phi$. - The link to the github repository in footnote 6 breaks the anonymity of submission rule as it reveals the identity of the some of the authors. **References** [1] Chen, Ricky TQ, et al. "Neural ordinary differential equations." Advances in neural information processing systems 31 (2018).

Rating

5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

2 fair

Presentation

2 fair

Contribution

3 good

Limitations

The authors adquately discuss limitations and future work.

Summary

The presented manuscript suggests a novel algorithm for learning dynamical systems trajectories called Koopman Kernel Regression. This method based on the Koopman operator theory proposes a unified way to construct observables such that the transformed dynamical system becomes LTI. Moreover, this study presents convergence results and the generalization risk bounds for the KKR method. The experiments confirm the given theory and demonstrate the performance of the KKR over existing kernel-based methods to learn the Koopman operator based on the observed snapshots.

Strengths

1) convergence results and generalization risk bounds are developed and supported by experimental results 2) also experiments illustrate the more accurate and robust w.r.t. prediction horizon compared to EDMD 3) the introductory part with a description of the Koopman operator approach and the corresponding challenges is well-written and clearly states the objective of this study 4) kernel for Koopman eigenfunction is derived and theoretically analyzed

Weaknesses

1) the key difference with the EDMD approach is not explicitly stated in the text, only the issues of the latter 2) it is unclear how to compose the complete algorithm with data samples as input and prediction as output from the presented theoretical results. The readability and accents in sections 3.2-3.3 should be fixed (from theory to practical usage) 3) how to select the kernel $k$ is not discussed in sufficient detail, only required properties are mentioned. 4) the discussion of the sampling strategy's importance for the obtained good experimental results is missing. In addition, how this step affects the performance of the KRR for arbitrary input data is interesting.

Questions

1) what is a particular form of the universal kernel $k$ used in experiments? Do EDMD and KKR use the same kernel inside? 2) what about the computational complexity of the proposed algorithm? How does it scale with the increasing dimension of the ODE that generates training trajectories? Experiments use only 1d trajectories. 3) please provide a complete step-by-step description of the proposed method in the algorithm LaTeX environment. It is hard to extract the final algorithm steps from the presented list of theorems and remarks. 4) the experiment section should be extended to more challenging dynamical systems like the Lorenz system, and systems with high-dimensional states. The runtime comparison with EDMD is very interesting, too. 5) please provide a more detailed description of the experimental setup. In particular, what spectrum sampling strategy is used and why? What happened if one will use another strategy and how to select the best one?

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

2 fair

Contribution

3 good

Limitations

No limitations are provided in the manuscript.

Summary

Learning time-series obeying nonlinear dynamics is essential for forecasting. Often Koopman operator theory is used to replace the non-linear dynamics with a linear system where the state evolution is given by a high-dimensional matrix that has to be estimated/learned. In this paper, the authors bring the kernel methods together with Koopman operators to obtain a mathematical framework where learning guarantees can be studied. Eigenfunctions of the Koopman operator associated to a dynamical system can be approximated by the basis functions of a Reproducing Kernel Hilbert Space (RKHS) and learning these eigenfunctions reduces to functional kernel regression. The paper derives risk bounds using statistical learning theory and demonstrate their method with experiments.

Strengths

The paper is well-written with a clear presentation. The problem of Koopman linearization is well-defined and techniques introduced are novel in terms of applying kernel methods for function approximation in this framework. Also, supplementary material is well-written and very helpful! Theorem 1 clearly defines the operator-valued Koopman Kernel for a single eigenfunction in terms of a chosen base kernel (e.g. Gaussian RBF) and the rank of the full Koopman kernel is obtained by direct-summation of $\bar D$ such RKHS's. The main problem in this formalism is the choice of eigenvalues for the Koopman Kernel. The authors propose a method for sampling eigenvalues, while it is not clear how to me (See weaknesses).

Weaknesses

1. In my opinion, previous methods are not explained well, and it is not easy to understand how this method differs from the previous works. 2. It could be helpful to comment on the spectral biases of the Koopman Kernels based on the base kernel and also the choice of Koopman eigenvalues. To me, it is still not clear how $\mu_i$'s are chosen, and it is essential to understand that point because Koopman kernel requires both the choice of base kernel and the spectrum. 3. Experiments might be extended to cover how the kernel hyperparameters affect the regression risk. For example, it is well-known in kernel methods community that certain spectral decay is essential for generalization. Since there are two spectra involved in this case (spectrum of the base kernel and spectrum of Koopman operator), it would be interesting to see how both affect generalization.

Questions

1. What is the computational complexity/cost of this method, and how does it relate to the existing methods?

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

4 excellent

Presentation

4 excellent

Contribution

3 good

Limitations

Limitations section is missing.

Review to the rebuttal

Dear authors, Thanks for the detailed rebuttal. The additional experiments have solved my concerns. I am happy to amend my score of the paper.