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:Gromov-Wasserstein(GW)框架通过在这些点之间寻求一种低扭曲、几何式的配置,为这些问题提供了一个日益流行的答案。作为非曲线,最佳运输(OT)的四面形概括,GW是硬的。虽然实践者常常将GW大约作为嵌入的固定的OT问题序列来解决GW,但这一方法的立方复杂性(以样品价格计)是一个障碍。我们在此工作中展示了将可接受组合的组合限制于低等级因素化的OT问题的最新变种是如何非常适合GW的解决方案的:在应用到GW时,我们表明这一方法不仅能够将GW问题作为一个固定的固定点,而且能够从实际成本基数的基数(以美元计)中得益益。