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On the Effect of Quantization on Dynamic Mode Decomposition

Dipankar Maity, Debdipta Goswami, Sriram Narayanan

TL;DR

This work studies how quantization affects Dynamic Mode Decomposition and EDMD for estimating the Koopman operator. Using dither quantization, it shows that in the large-data limit the quantized EDMD optimization is equivalent to a regularized EDMD with regularization parameter $\gamma = \epsilon^2/12$, enabling recovery of the unquantized estimate as $\epsilon \to 0$. In the finite-data regime, the quantized estimator deviates from the unquantized one by a perturbation $\mathcal{K}_{\varepsilon}$ with $\|\mathcal{K}_{\varepsilon}\| = O(\epsilon)$, and the deviation vanishes as quantization becomes finer. Numerical experiments on a pendulum with negative damping, a Van der Pol oscillator, and flow past a cylinder corroborate the theory, showing exponential improvement of prediction accuracy with increased word length and demonstrating the practical viability of quantized DMD. Collectively, the results offer a principled framework for performing Koopman-based analysis under communication and resource constraints and suggest pathways to recover unquantized estimates from quantized data.

Abstract

Dynamic Mode Decomposition (DMD) is a widely used data-driven algorithm for estimating the Koopman Operator.This paper investigates how the estimation process is affected when the data is quantized. Specifically, we examine the fundamental connection between estimates of the operator obtained from unquantized data and those from quantized data. Furthermore, using the law of large numbers, we demonstrate that, under a large data regime, the quantized estimate can be considered a regularized version of the unquantized estimate. This key theoretical finding paves the way to accurately recover the unquantized estimate from quantized data. We also explore the relationship between the two estimates in the finite data regime. The theory is validated through repeated numerical experiments conducted on three different dynamical systems.

On the Effect of Quantization on Dynamic Mode Decomposition

TL;DR

This work studies how quantization affects Dynamic Mode Decomposition and EDMD for estimating the Koopman operator. Using dither quantization, it shows that in the large-data limit the quantized EDMD optimization is equivalent to a regularized EDMD with regularization parameter , enabling recovery of the unquantized estimate as . In the finite-data regime, the quantized estimator deviates from the unquantized one by a perturbation with , and the deviation vanishes as quantization becomes finer. Numerical experiments on a pendulum with negative damping, a Van der Pol oscillator, and flow past a cylinder corroborate the theory, showing exponential improvement of prediction accuracy with increased word length and demonstrating the practical viability of quantized DMD. Collectively, the results offer a principled framework for performing Koopman-based analysis under communication and resource constraints and suggest pathways to recover unquantized estimates from quantized data.

Abstract

Dynamic Mode Decomposition (DMD) is a widely used data-driven algorithm for estimating the Koopman Operator.This paper investigates how the estimation process is affected when the data is quantized. Specifically, we examine the fundamental connection between estimates of the operator obtained from unquantized data and those from quantized data. Furthermore, using the law of large numbers, we demonstrate that, under a large data regime, the quantized estimate can be considered a regularized version of the unquantized estimate. This key theoretical finding paves the way to accurately recover the unquantized estimate from quantized data. We also explore the relationship between the two estimates in the finite data regime. The theory is validated through repeated numerical experiments conducted on three different dynamical systems.
Paper Structure (20 sections, 2 theorems, 48 equations, 4 figures)

This paper contains 20 sections, 2 theorems, 48 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Koopman Operator Theory
  4. Approximation of Koopman Operator: Dynamic Mode Decomposition
  5. Dither Quantization
  6. Problem Statement
  7. DMD with Quantized Data
  8. Finite Data Regime
  9. Numerical Examples
  10. Pendulum with negative damping
  11. Van der Pol oscillator
  12. Flow past cylinder
  13. Discussions
  14. Regularized DMD
  15. Quality vs. Quantity Trade-off
  16. ...and 5 more sections

Key Result

Theorem 1

For a large $T$,

Figures (4)

  • Figure 1: Error profile for negatively-damped pendulum \ref{['Eq: Pendulum']}.
  • Figure 2: Error profile for Van der Pol oscillator \ref{['Eq: Vanderpol']}.
  • Figure 3: Error profile for flow past cylinder.
  • Figure 4: Snapshot of velocity field at $t=80$s.

Theorems & Definitions (9)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof