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Gromov-Wasserstein Problem with Cyclic Symmetry

We propose novel fast algorithms for the Gromov--Wasserstein problem (GW) with cyclic symmetry of input data. This problem naturally appears as an object-matching task, which underlies various real-world computer vision applications, e.g., image registration, point cloud registration, stereo matching, and 3D reconstruction. Gradient-based algorithms have been widely used to solve GW, and our main idea is to utilize the following remarkable property that emerges in GW with cyclic symmetry: By setting the initial solution to have cyclic symmetry, all intermediate solutions and matrices that appear in the gradient-based algorithms have the same cyclic symmetry until convergence. Based on this property, our gradient-based algorithms restrict the solution space to have cyclic symmetry and update only one symmetric part of solutions and matrices at each iteration, resulting in faster computation. Moreover, our algorithms solve the optimal transport problem at each iteration, which also exhibits cyclic symmetry. This problem can be solved efficiently, and as a result, our algorithms perform significantly faster. Experiments showed the effectiveness of our algorithms in synthetic and real-world data with strict and approximate cyclic symmetry.

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