Gromov-Wasserstein Transfer Operators

TL;DR

This paper introduces a method to estimate dynamical systems using Gromov-Wasserstein transport plans, leveraging entropy regularization and spectral clustering to identify coherent structures in data.

Contribution

It proposes a novel approach combining GW transport with transfer operators for dynamical system estimation, including variants for labeled and noisy data.

Findings

Effective extraction of coherent structures demonstrated on numerical examples.

Utilization of Sinkhorn algorithm for efficient computation.

Quantitative assessment of shape-coherence achieved.

Abstract

Gromov-Wasserstein (GW) transport is inherently invariant under isometric transformations of the data. Having this property in mind, we propose to estimate dynamical systems by transfer operators derived from GW transport plans, when merely the initial and final states are known. We focus on entropy regularized GW transport, which allows to utilize the fast Sinkhorn algorithm and a spectral clustering procedure to extract coherent structures. Moreover, the GW framework provides a natural quantitative assessment on the shape-coherence of the extracted structures. We discuss fused and unbalanced variants of GW transport for labelled and noisy data, respectively. Our models are verified by three numerical examples of dynamical systems with governing rotational forces.