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This paper presents a framework for computing the Gromov-Wasserstein problembetween two sets of points in low dimensional spaces, where the discrepancy isthe squared Euclidean norm. The Gromov-Wasserstein problem is a generalizationof the optimal transport problem that finds the assignment between two setspreserving pairwise distances as much as possible. This can be used to quantifythe similarity between two formations or shapes, a common problem in AI andmachine learning. The problem can be formulated as a Quadratic AssignmentProblem (QAP), which is in general computationally intractable even for smallproblems. Our framework addresses this challenge by reformulating the QAP as anoptimization problem with a low-dimensional domain, leveraging the fact thatthe problem can be expressed as a concave quadratic optimization problem withlow rank. The method scales well with the number of points, and it can be usedto find the global solution for large-scale problems with thousands of points.We compare the computational complexity of our approach with state-of-the-artmethods on synthetic problems and apply it to a near-symmetrical problem whichis of particular interest in computational biology.

Globally solving the Gromov-Wasserstein problem for point clouds in low dimensional Euclidean spaces