KoopSTD: Reliable Similarity Analysis between Dynamical Systems via Approximating Koopman Spectrum with Timescale Decoupling

We sincerely appreciate your thorough assessment of our work. We are delighted that you recognize KoopSTD's contributions to providing a robust and multi-scale similarity metric for nonlinear dynamical systems. In particular, we deeply appreciate your time and careful attention in examining our experimental setup, as well as the discussion of its theoretical foundations and empirical validation. Your recognition that our results convincingly support our claims is especially encouraging.

Thank you again for your time and evaluation!

Thanks for your careful review and comments!

Glad to see that you deem our work interesting and have a promising impact on many fields. Please kindly find our point-to-point responses below.

Concerns

Q1: Presenting the invariance property as 'theorem'.

We revise the invariance property from "Theorem” to "Proposition”.

Q2: Using "theoretical guarantees" to describe the properties of KoopSTD.

We revise the "theoretical guarantees" to "theoretical groundedness" accordingly.

Q3: Improper usage of "Koopman matrix" and "$𝐗_{t+1}=𝐀_{1}𝐗_{t}$".

We correct the terminology to "approximated Koopman operator" and "$g(𝐗_{t+1})=𝐀_{1}(g(𝐗_{t}))$", respectively.

Q4: Incorrect equality $d(cλ(𝐀_1),cλ(𝐀_2))=d(λ(𝐀_1),λ(𝐀_2))$.

We fix this error by removing line 250 (left) in the main manuscript to let $d(c𝓕_1,c𝓕_2)=d(λ(𝐀_1),λ(𝐀_2))$ directly.

Q5: Unclear proof of three invariance properties.

We refine our proof by unifying three invariance properties into a single proposition below.

Proposition. Let $𝐗_1[t+1]= 𝓕_1({𝐗_1[t]})$ and $𝐗_2[t+1]= 𝓕_2({𝐗_2[t]})$ be two time-discrete dynamical systems, where $𝓕_1, 𝓕_2 : ℝ^{N_d} \to ℝ^{N_d}$. The dissimilarity $d(𝓕_1, 𝓕_2)$, as measured by KoopSTD, remains invariant under invertible linear transformations 𝓣,

$$d(𝓣(𝓕_1, 𝓕_2)) = d(𝓕_1, 𝓕_2),$$

where $𝓣=${$𝐗 ↦ 𝐗𝐐: 𝐐 \in GL(N_d, ℝ)$}.

Proof. For a dynamical system $𝐗[t+1]= 𝓕({𝐗[t]})$. The KoopSTD seeks to obtain the approximate Koopman operator $𝐀$ by firstly mapping $𝐗\inℝ^{T\times N_d}$ into a time-frequency representation $𝐙\inℝ^{(\frac{T-l}{s}+1)\times(\frac{l}{2}+1)\times N_d}$ using STFT.

By the linearity of the STFT, we can express the process of STFT as:

$$𝐙=\text{STFT}(𝐗)=𝓛_{tf}𝐗,$$

where $𝓛_{tf}$ refers to the linear operator of the STFT.

Then, KoopSTD computes the right singular vectors $𝑽$ of $𝐙$ by solving SVD:

$$𝐙^* 𝐙=(𝑽\Sigma 𝑼^*)(𝑼\Sigma 𝑽^*)=𝑽\Sigma^2𝑽^*.$$

Therefore, the Koopman operator can be approximated based on temporal snapshots $𝐖_{𝑽}$ of $𝑽$:

$$𝐀 = 𝐖_𝑽'𝐖_𝑽^{†}.$$

Given any transformed input $𝐗𝐐$, we can also derive its time-frequency representation $𝐙_{𝓣}$:

$$𝐙_{𝓣}=\text{STFT}(𝐗𝐐)=𝓛_{tf}𝐗𝐐=𝐙𝐐.$$

Then, the transformed singular vectors $𝑽_{𝓣}$ of $𝐙_{𝓣}$ can be derived as follows:

$$𝐙_{𝓣}^* 𝐙_{𝓣}=(𝐙𝐐)^* 𝐙𝐐=𝐐^* 𝐙^* 𝐙𝐐 =(𝐐^* 𝑽)\Sigma^2(𝑽^* 𝐐),$$ $$𝑽_{𝓣}=𝐐^*𝑽.$$

The transformed snapshots $𝐖_{𝑽,𝓣}$ of $𝑽_{𝓣}$ can also be derived:

$$𝐖_{𝑽,𝓣}=𝐐^*𝐖_{𝑽},$$ $$𝐖'_{𝑽,𝓣}=𝐐^*𝐖'_𝑽.$$

Thus, the transformed approximated Koopman operator $𝐀_{𝓣}$ can be denoted as follows:

$$𝐀_{𝓣}=𝐖_{𝑽,𝓣}'𝐖_{𝑽,𝓣}^{†}=𝐐^* 𝐀(𝐐^*)^{-1}.$$

By definition of similarity transformation, the matrices $𝐀_{𝓣}$ and $𝐀$ are similar, implying that they share the same eigenvalues:

$$λ(𝐀_{𝓣})=λ(𝐀).$$

Based on Definition (1) and the above derivation, we have:

$$d(𝓣(𝓕_{1},𝓕_{2}))= d(λ(𝐀_{1,𝓣}),λ(𝐀_{2,𝓣}))=d(λ(𝐀_{1}), λ(𝐀_{2})) = d(𝓕_1,𝓕_2).$$

Since the following transformations on the feature dimension are also invertible linear transformations, we can deduce that KoopSTD remains invariant to the following transformations,

• Isotropic scaling$:𝓣_{IS}=${$𝐗↦ 𝐗𝐐:𝐐=qI_{N_d},q\in ℝ^+$}.
• Rotations$:𝓣_{R}=${$𝐗 ↦ 𝐗𝐐:𝐐 \in O(N_d)$}, where $O(N_d) :=$ {$ℝ^{N_d \times N_d}, 𝐐^T𝐐=I_{N_d}$} denote the orthogonal group.
• Permutations$:𝓣_{P}=${$𝐗 ↦ 𝐗𝐐_{\pi}: \pi \in P(N_d)$}, where $P(N_d)$ is the set of permutations on {$1,...,N_d$}.

Q6: The properties of KoopSTD stem from the invariance of the characteristic polynomial under similarity transformations. Does KoopSTD remain invariant to all similarity transformations?

As shown in our response to Q5, applying any invertible linear transformation to $𝐗$ is equivalent to applying a similarity transformation on the approximated Koopman operator $𝐀$, thereby ensuring KoopSTD's measurement remains invariant to all invertible linear transformations.

We emphasized the invariance properties in terms of isotropic scaling, rotations and permutations in the manuscript because they are essential for developing a reliable metric that ensures robust similarity measurements across different systems.

Related work of spectral error estimation provided by the reviewer

Thank you for providing insightful related work!

We agree that the performance of spectral error estimation using the Galerkin method can be influenced by the choice of basis. We have carefully read these relevant studies. The Introduction section will cover them briefly. Additionally, we will cite some of them in the Conclusion section as future work in effectively extracting the reliable modes.

Typos pointed out by the reviewer

Thanks for your careful review! We have proofread the manuscript, the typos will be fixed accordingly.

We sincerely appreciate the reviewer's valuable feedback!

Please kindly find the point-to-point response to the questions and suggestions.

Q1: How about applying DMD on the complex part of STFT results instead of the real part?

We appreciate the reviewer’s insightful question. We agree that discarding phase information will enhance robustness, as phase is highly sensitive to time shifts and minor perturbations. This sensitivity is particularly problematic in fMRI and other biological signals, where recording conditions and preprocessing can introduce unavoidable phase shifts.

From the perspective of the Koopman operator theory, the magnitude, which represents the signal's energy distribution, generally evolves in a smoother manner and is more amenable to linear approximation, making it more suitable for DMD. On the other hand, phase information is often linked to transient and localized changes, which are more challenging to capture within the linear framework of the Koopman operator theory. Therefore, in the KoopSTD framework, the real part is more suitable.

We acknowledge that incorporating phase information in a controlled manner to enrich the description of system dynamics could be an interesting future direction. We will add a discussion on this point in the Conclusion section in the revised manuscript.

Once again, we sincerely thank the reviewer for bringing this to our attention.

Q2: How can we determine the hyperparameters of KoopSTD in practice, including the number of modes?

Thanks for pointing out this question! We agree that the choice of hyperparameters — particularly the STFT window length $l$ and the number of preserved modes $r$ — deserves more discussion. We will add a brief description in the Experimental Results section in the revised main manuscript and a detailed discussion in the revised appendix.

In most DMD-related work, hyperparameters are typically chosen heuristically, relying on prior knowledge and experience with the specific system’s characteristics[1,2,3]. In contrast, by conducting an additional sensitivity analysis, we demonstrate that our KoopSTD remains robust across a wide range of these parameters, reducing the reliance on manual tuning. Specifically, we conducted the experiments on the classical Lorenz system, measuring the silhouette coefficients of the distance matrices obtained by KoopSTD with various $<l,r>$ combinations. In this setting, the silhouette coefficient$\in [-1,1]$ is the larger, the better.

The results are summarized in the table below. It is noteworthy that the results in most of the configurations are greater than 0.9. This indicates that KoopSTD is able to consistently distinguish different dynamical behaviors, even with diverse hyperparameter settings, highlighting its robustness in practice without requiring extensive hyperparameter tuning.

Note that increasing $r$ within a reasonable range preserves more meaningful modes, potentially enhancing the representation of the system's dynamic patterns. However, continuously increasing $r$ presents two key drawbacks:

1. Firstly, an excessively large number of modes can introduce uncontrollable spectral error, reducing the accuracy of the Koopman approximation.
2. Secondly, the computational cost can significantly rise. While the computational cost between a single pair comparison remains manageable, comparing all pairs in a large dataset becomes substantially more time-consuming.

[1] Kutz et al., Multiresolution dynamic mode decomposition. In SIAM Journal on Applied Dynamical Systems, 2016.

[2] Noack et al., Recursive dynamic mode decomposition of transient and post-transient wake flows. In Journal of Fluid Mechanics, 2016.

[3] Brunto et al., Chaos as an intermittently forced linear system. In Nature communications, 2017.