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On the Effect of Quantization on Extended Dynamic Mode Decomposition

Dipankar Maity, Debdipta Goswami

TL;DR

This paper investigates how the estimation process is affected when the data is quantized, and demonstrates that, under a large data regime, the quantized estimate can be considered a regularized version of the unquantized estimate.

Abstract

Extended Dynamic Mode Decomposition (EDMD) is a widely used data-driven algorithm for estimating the Koopman Operator. EDMD extends Dynamic Mode Decomposition (DMD) by lifting the snapshot data using nonlinear dictionary functions before performing the estimation. This letter 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 via EDMD. 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. We also explore the relationship between the two estimates in the finite data regime. We further analyze the effect of nonlinear lifting functions on this regularization due to quantization. The theory is validated through repeated numerical experiments conducted on two different dynamical systems.

On the Effect of Quantization on Extended Dynamic Mode Decomposition

TL;DR

This paper investigates how the estimation process is affected when the data is quantized, and demonstrates that, under a large data regime, the quantized estimate can be considered a regularized version of the unquantized estimate.

Abstract

Extended Dynamic Mode Decomposition (EDMD) is a widely used data-driven algorithm for estimating the Koopman Operator. EDMD extends Dynamic Mode Decomposition (DMD) by lifting the snapshot data using nonlinear dictionary functions before performing the estimation. This letter 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 via EDMD. 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. We also explore the relationship between the two estimates in the finite data regime. We further analyze the effect of nonlinear lifting functions on this regularization due to quantization. The theory is validated through repeated numerical experiments conducted on two different dynamical systems.
Paper Structure (15 sections, 2 theorems, 44 equations, 2 figures)

This paper contains 15 sections, 2 theorems, 44 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Koopman Operator Theory
  4. Approximation of Koopman Operator via EDMD
  5. Dither Quantization
  6. Problem Statement
  7. EDMD with Quantized Data
  8. Large Data regime
  9. Regularized $\texttt{DQ-EDMD}$
  10. Finite Data Regime
  11. Numerical Examples
  12. Pendulum with negative damping
  13. Van der Pol oscillator
  14. Conclusions
  15. Proof of \ref{['thm:equivalence']}

Key Result

Theorem 1

As $T \to \infty$ and $|\epsilon|<1$, $\tilde{\mathcal{K}}_{\rm EDMD}$ converges almost surely to the solution of the following regularized EDMD where $\beta(\epsilon)$ and $\Gamma(\epsilon)$ are $O(\epsilon^2)$, and $O(\cdot)$ is the Big-O notation.

Figures (2)

  • Figure 1: Error and prediction profile for negatively-damped pendulum \ref{['Eq: Pendulum']}.
  • Figure 2: Error and prediction profile for Van der Pol oscilator \ref{['Eq: Vanderpol']}.

Theorems & Definitions (5)

  • Theorem 1: Large data regime result
  • proof
  • proof
  • Theorem 2: Finite data regime result
  • proof