Summary
This paper develops bounds on how close the approximated Koopman modes and eigenvalues are to the true eigenvalues and modes, for two important classes of methods used to compute the Koopman mode decomposition. The authors find that one class, Principal Component Regression, which included extended dynamic mode decomposition (EDMD), can suffer more from poorly chosen kernels and can have larger bias than Reduced Rank Regression (RRR). They additionally provide an empirical method for determining spurious eigenvalues, which can be used for model selection.
Strengths
1. This paper is well written and easy to follow.
2. This paper provides new techniques for computing bounds on the approximated Koopman spectral objects, and the discovery that PCR can have larger bias than RRR, are important ones for the field.
3. This paper provides a new empirical method for identifying spurious eigenvalues and making model selection. Again, both of these are important topics for the field and for the application of numerical methods in applied settings.
4. The numerical examples provided in Figs. 1-3 are helpful for understanding the theory developed, and provide support for the developed claims.
Weaknesses
1. Klus et al., 2016 and Korda and Mezic, 2018, as examples, proved the convergence of EDMD to the true Koopman operator, when $M \rightarrow \infty$, where $M$ is the number of data points. This work seems to be missing from the paper, as does discussion surrounding how the paper differs from that work. I assume the primary difference is that this paper's results are not in the asymptotic limit (although, they are in the sense that the bounds for RRR, for example, in the Gaussian case converge as $1/\sqrt{n}$, which approaches $0$ as $n\rightarrow \infty$). Additionally, the results for PCR obtained by this paper would suggest that in the asymptotic limit EDMD does not converge, since it has a bias. Discussion on how this is reconciled with the work of Klus et al., 2016 and Korda and Mezic, 2018, is necessary.
2. Fig. 3 was confusing. Was the best estimator found on the test data set, and then the red line in Fig. 3 the result of applying it to the validation data? A secondary panel in that figure describing what was being done would be helpful.
MINOR COMMENTS:
1. It was unclear to me how $| \lambda_i - \mu_{j(i)} | \leq || (A_\pi - \lambda_i I)^{-1}||^{-1}$ leads to observing that $||(A_\pi - S\hat{G})\hat{\psi}_i|| \leq \mathcal{E}(\hat{G})\eta(\hat{\psi})$ (lines 154-155). Adding a little more detail/comment on this would be helpful.
2. "left hand side" (line 156) should be "right hand side" no?
3. The connection between DMD and KMD should be discussed (lines 22-25) (Rowley et al., 2009).
4. The original EDMD paper (Williams et al., 2015) should be cited when discussing EDMD for the first time (line 31).
5. Very minor but both "non-linear" and "nonlinear" are written.
Questions
1. How does this work compare to previous work studying the convergence of EDMD (e.g., Klus et al., 2016; Korda and Mezic, 2018)?
2. What exactly is Fig. 2 showing (in terms of details)?
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.
Limitations
The authors did a good job being clear that their work was limited to self-adjoint operators.