Globally solving the Gromov-Wasserstein problem for point clouds in low dimensional Euclidean spaces
This provides a scalable method for quantifying similarity between shapes or formations, which is incremental but addresses a known bottleneck in AI and machine learning applications like computational biology.
The paper tackles the computationally intractable Gromov-Wasserstein problem for point clouds in low-dimensional Euclidean spaces by reformulating it as a concave quadratic optimization problem with low rank, enabling global solutions for large-scale problems with thousands of points.
This paper presents a framework for computing the Gromov-Wasserstein problem between two sets of points in low dimensional spaces, where the discrepancy is the squared Euclidean norm. The Gromov-Wasserstein problem is a generalization of the optimal transport problem that finds the assignment between two sets preserving pairwise distances as much as possible. This can be used to quantify the similarity between two formations or shapes, a common problem in AI and machine learning. The problem can be formulated as a Quadratic Assignment Problem (QAP), which is in general computationally intractable even for small problems. Our framework addresses this challenge by reformulating the QAP as an optimization problem with a low-dimensional domain, leveraging the fact that the problem can be expressed as a concave quadratic optimization problem with low rank. The method scales well with the number of points, and it can be used to find the global solution for large-scale problems with thousands of points. We compare the computational complexity of our approach with state-of-the-art methods on synthetic problems and apply it to a near-symmetrical problem which is of particular interest in computational biology.