@inproceedings{
chen2024semidefinite,
title={Semidefinite Relaxations of the Gromov-Wasserstein Distance},
author={Junyu Chen and Binh Nguyen and Shang Hui Koh and Yong Sheng Soh},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=rM3FFH1mqk}
}
The Gromov-Wasserstein (GW) distance is an extension of the optimal transport problem that allows one to match objects between incomparable spaces.  At its core, the GW distance is specified as the solution of a non-convex quadratic program and is not known to be tractable to solve.  In particular, existing solvers for the GW distance are only able to find locally optimal solutions.  In this work, we propose a semi-definite programming (SDP) relaxation of the GW distance. The relaxation can be viewed as the Lagrangian dual of the GW distance augmented with constraints that relate to the linear and quadratic terms of transportation plans. In particular, our relaxation provides a tractable (polynomial-time) algorithm to compute globally optimal transportation plans (in some instances) together with an accompanying proof of global optimality.  Our numerical experiments suggest that the proposed relaxation is strong in that it frequently computes the globally optimal solution.  Our Python implementation is available at https://github.com/tbng/gwsdp.
Semidefinite Relaxations of the Gromov-Wasserstein Distance
Junyu Chen Binh Nguyen Shang Hui Koh Yong Sheng Soh