A fundamental challenge in data science is to match disparate point sets with each other. While optimal transport efficiently minimizes point displacements under a bijectivity constraint, it is inherently sensitive to rotations. Conversely, minimizing distortions via the Gromov-Wasserstein (GW) framework addresses this limitation but introduces a non-convex, computationally demanding optimization problem. In this work, we identify a broad class of distortion penalties that reduce to a simple alignment problem within a lifted feature space. Leveraging this insight, we introduce an iterative GW solver with a linear memory footprint and quadratic (rather than cubic) time complexity. Our method is differentiable, comes with strong theoretical guarantees, and scales to hundreds of thousands of points in minutes. This efficiency unlocks a wide range of geometric applications and enables the exploration of the GW energy landscape, whose local minima encode the symmetries of the matching problem.

翻译：数据科学中的一项基本挑战是将不同的点集相互匹配。虽然最优传输在双射约束下能高效地最小化位移，但其本质上对旋转敏感。相反，通过Gromov-Wasserstein (GW) 框架最小化形变解决了这一局限，却引入了非凸且计算量大的优化问题。在本研究中，我们识别出一类广泛的形变惩罚项，它们可归结为提升特征空间中的简单对齐问题。基于这一发现，我们提出了一种迭代式GW求解器，其内存占用呈线性，时间复杂度为二次方（而非三次方）。该方法具有可微性、强理论保证，并能在数分钟内扩展到数十万个点。这种高效性解锁了广泛的几何应用，并使得探索GW能量景观成为可能——其局部最小值编码了匹配问题的对称性。