Randomized Methods for Kernelized DMD

TL;DR

This paper introduces a randomized algorithm using RPCholesky for kernelized DMD, significantly improving computational efficiency and stability for large-scale data analysis in dynamical systems.

Contribution

It proposes a novel application of the RPCholesky algorithm to kernelized DMD, enhancing stability and scalability through adaptive randomized sampling.

Findings

Improved stability guarantees over previous methods

Enhanced computational efficiency for large datasets

Successful application to high-dimensional benchmark problems

Abstract

Dynamic Mode Decomposition (DMD) is a data-driven method related to Koopman operator theory that extracts information about dominant dynamics from data snapshots. In this paper we examine techniques to accelerate the application of DMD to large-scale data sets with an eye on randomized techniques. Randomized techniques exploit low-rank matrix approximations at a much smaller computational cost, therefore permitting the use of increased data set sizes. In particular, we propose the application of the RPCholesky algorithm in the setting of kernelized DMD (KDMD). This algorithm relies on adaptive randomized sampling to approximate positive semidefinite kernel matrices and provides better stability guarantees than previously implemented randomized methods for KDMD. Differences between existing competitive randomized techniques and our proposed implementation are discussed with a focus on numerical stability and tradeoff between exploration and exploitation of information obtained from data. The efficacy of this new combination of algorithms is demonstrated on well-established benchmark problems from DMD literature increasing in problem dimension.