On the Effect of Quantization on Extended Dynamic Mode Decomposition

TL;DR

This paper studies how quantizing data impacts the estimation of the Koopman Operator using EDMD, revealing a regularization effect in large data regimes and analyzing the influence of nonlinear lifting functions.

Contribution

It establishes a theoretical connection between quantized and unquantized EDMD estimates, showing quantization acts as regularization, supported by numerical experiments.

Findings

Quantization introduces a regularization effect in EDMD estimates.

Large data regimes mitigate quantization effects, improving estimate accuracy.

Nonlinear lifting functions influence the regularization due to quantization.

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.