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：数据科学中的一个基础性挑战是如何匹配不同的点集。最优传输方法在双射约束下能有效最小化点位移，但其本质上对旋转敏感。相反，通过Gromov-Wasserstein（GW）框架最小化畸变虽能解决这一局限，却引入了非凸且计算复杂的优化问题。本研究识别出一类广泛的畸变惩罚函数，它们可转化为提升特征空间中的简单对齐问题。基于这一发现，我们提出了一种具有线性内存占用和二次（而非三次）时间复杂度的迭代式GW求解器。该方法具有可微性，具备坚实的理论保证，并能在数分钟内处理数十万个数据点。这种高效性为几何应用开辟了广阔前景，并使得探索GW能量景观成为可能——该景观的局部极小值编码了匹配问题的对称性信息。