The ability to align points across two related yet incomparable point clouds (e.g. living in different spaces) plays an important role in machine learning. The Gromov-Wasserstein (GW) framework provides an increasingly popular answer to such problems, by seeking a low-distortion, geometry-preserving assignment between these points. As a non-convex, quadratic generalization of optimal transport (OT), GW is NP-hard. While practitioners often resort to solving GW approximately as a nested sequence of entropy-regularized OT problems, the cubic complexity (in the number $n$ of samples) of that approach is a roadblock. We show in this work how a recent variant of the OT problem that restricts the set of admissible couplings to those having a low-rank factorization is remarkably well suited to the resolution of GW: when applied to GW, we show that this approach is not only able to compute a stationary point of the GW problem in time $O(n^2)$, but also uniquely positioned to benefit from the knowledge that the initial cost matrices are low-rank, to yield a linear time $O(n)$ GW approximation. Our approach yields similar results, yet orders of magnitude faster computation than the SoTA entropic GW approaches, on both simulated and real data.

翻译：在两个相关但不可比的点云（例如，位于不同空间中的点云）之间对齐点的能力在机器学习中扮演着重要角色。Gromov-Wasserstein（GW）框架通过寻找点间低失真、保持几何结构的指派，为这类问题提供了日益流行的解决方案。作为最优传输（OT）的非凸、二次推广，GW是NP难的。尽管实践者常通过将GW近似求解为嵌套序列的熵正则化OT问题，但该方法的三次复杂度（关于样本数$n$）成为瓶颈。本文中我们证明，OT问题的一个最新变体——将容许耦合集限制为具有低秩分解的形式——异常适合用于求解GW：当应用于GW时，该方法不仅能在$O(n^2)$时间内计算GW问题的驻点，还能独特地利用初始代价矩阵为低秩的先验知识，从而得到线性时间$O(n)$的GW近似。我们的方法在模拟数据和真实数据上均能产生与最先进熵正则化GW方法相似的结果，但计算速度却快数个数量级。