A Provably Convergent and Practical Algorithm for Gromov--Wasserstein Optimal Transport

TL;DR

This paper introduces a new inexact projected-gradient algorithm for Gromov--Wasserstein optimal transport that guarantees convergence and is practical for large-scale problems.

Contribution

It proposes a verifiable inexact projection condition that ensures convergence, bridging the gap between practical implementations and theoretical guarantees.

Findings

Proposed method converges to stationary points under verifiable conditions.

The algorithm maintains simplicity and scalability for large GWOT problems.

Provides rigorous convergence guarantees for practical projected-gradient schemes.

Abstract

Gromov--Wasserstein optimal transport (GWOT) aligns metric measure spaces by matching their within-domain relational structures, but large-scale GWOT remains challenging because its objective is nonconvex and projection onto the transport polytope is often solved only approximately in practice. This leads to a gap between practical projected-gradient implementations and convergence theory, which typically assumes exact projections. For squared-loss GWOT, we propose an inexact projected-gradient framework with a verifiable feasibility-residual-based inexact condition for the projection subproblem. This condition is directly computable and avoids unknown quantities such as the exact projection point. Under this implementable condition, we prove subsequential convergence to stationary points and, with a mild tolerance-decay condition, convergence of the whole sequence. The resulting method retains the simplicity and sparsity of projected-gradient schemes while providing rigorous convergence guarantees, turning projected-gradient methods into a principled and scalable approach for GWOT with provable reliability.