insight - Dynamical systems analysis - # Quantization impact on Koopman operator estimation using Dynamic Mode Decomposition

Core Concepts

The core message of this work is that the estimation of the Koopman operator using Dynamic Mode Decomposition (DMD) with quantized data can be interpreted as a regularized DMD problem with unquantized data, where the regularization parameter depends on the quantization resolution. This connection provides a framework to potentially recover the unquantized Koopman operator estimate from the quantized data.

Abstract

The authors investigate the effects of dither quantization on the Extended Dynamic Mode Decomposition (EDMD) method for estimating the Koopman operator. The key insights are:
In the large data regime, the EDMD optimization problem with quantized data can be interpreted as a regularized EDMD optimization with unquantized data, where the regularization parameter depends on the quantization resolution. This implies that the quantized estimate converges to the unquantized estimate as the quantization resolution improves.
In the finite data regime, the authors show that the quantized estimate can be expressed as the sum of the unquantized estimate and an error term that scales linearly with the quantization resolution. This provides an analytical characterization of the difference between the quantized and unquantized estimates.
The authors validate the theoretical findings through numerical experiments on three different dynamical systems: a negatively-damped pendulum, a Van der Pol oscillator, and flow past a cylinder. The results demonstrate that the prediction error decreases exponentially with the quantization word length, indicating the effectiveness of dither quantization in preserving the Koopman operator estimation quality.
The authors also discuss the potential to recover the unquantized Koopman operator estimate from the quantized data by solving a regularized DMD problem, where the regularization parameter is chosen based on the quantization resolution.

Stats

The authors use the following key metrics to quantify the impact of quantization:
∥KDMD - ˜KDMD∥/∥KDMD∥: Normalized estimation error for the full-order Koopman operator.
∥KDMD,r - ˜KDMD,r∥/∥KDMD,r∥: Normalized estimation error for the reduced-order Koopman operator.
(1/T) Σ(t=0 to T-1) ∥x̂t - xt∥/∥xt∥: Time-average normalized prediction error using the reduced-order Koopman operator.

Quotes

"The core message of this work is that the estimation of the Koopman operator using Dynamic Mode Decomposition (DMD) with quantized data can be interpreted as a regularized DMD problem with unquantized data, where the regularization parameter depends on the quantization resolution."
"The authors validate the theoretical findings through numerical experiments on three different dynamical systems: a negatively-damped pendulum, a Van der Pol oscillator, and flow past a cylinder."

Key Insights Distilled From

by Dipankar Mai... at **arxiv.org** 04-03-2024

Deeper Inquiries

To further investigate the statistical properties of the error term Kϵ in the finite data regime as presented in Theorem 2, one could delve into the distributional characteristics of Kϵ. Analyzing the moments of Kϵ, such as its mean, variance, skewness, and kurtosis, would provide a deeper understanding of its behavior. Additionally, exploring the correlation structure of Kϵ with the quantization resolution ϵ and the underlying dynamics could offer valuable insights. Conducting Monte Carlo simulations with varying quantization resolutions and data lengths could help in observing the convergence properties of Kϵ and its relationship with the estimation error. Furthermore, investigating the spectral properties of Kϵ through eigenvalue analysis could reveal important information about its stability and convergence behavior.

While the proposed regularized DMD approach offers a promising framework for recovering the unquantized Koopman operator estimate from quantized data, there are potential limitations and drawbacks to consider. One limitation is the reliance on the regularization parameter γ, which needs to be carefully chosen to ensure the convexity of the optimization problem. Selecting an inappropriate value for γ could lead to suboptimal solutions or even non-convergence of the algorithm. Another drawback is the assumption of full row rank data matrices Φ and Φ′, which may not always hold in practical scenarios. In cases where the data matrices are rank deficient, the recovery of the unquantized estimate using the regularized approach may be challenging. Additionally, the computational complexity of solving the regularized DMD problem for large datasets could be a limiting factor, especially in real-time or resource-constrained applications.

The insights from this work on the effect of quantization on Dynamic Mode Decomposition (DMD) can be extended to other data-driven Koopman operator estimation techniques beyond DMD/EDMD, such as Koopman autoencoder architectures. By incorporating dither quantization into the training and inference processes of Koopman autoencoders, one can explore how quantization affects the learned representations and the accuracy of the Koopman operator estimation. Analyzing the impact of quantization on the latent space learned by Koopman autoencoders and its implications for system identification and prediction tasks would be a valuable extension. Furthermore, investigating the trade-offs between quantization resolution, model complexity, and estimation accuracy in the context of Koopman autoencoders could provide insights into optimizing these architectures for practical applications.

0