Note on computational complexity of the Gromov-Wasserstein distance

TL;DR

This paper analyzes the computational complexity of the Gromov-Wasserstein distance, revealing its inherently non-convex quadratic optimization structure and illustrating this with explicit examples.

Contribution

It clarifies the non-convex quadratic nature of the Gromov-Wasserstein distance optimization problem through detailed analysis and examples.

Findings

Gromov-Wasserstein distance optimization is non-convex and quadratic.

Explicit examples demonstrate the non-convexity.

Provides insights into the computational challenges of the distance.

Abstract

This note addresses computational difficulty of the Gromov-Wasserstein distance frequently mentioned in the literature. We provide details on the structure of the Gromov-Wasserstein distance optimization problem that show its non-convex quadratic nature for any instance of an input data. We further illustrate the non-convexity of the problem with several explicit examples.