Optimal Transport (OT) has proven to be a powerful tool to compare probability distributions in machine learning, but dealing with probability measures lying in different spaces remains an open problem. To address this issue, the Gromov Wasserstein distance (GW) only considers intra-distribution pairwise (dis)similarities. However, for two (discrete) distributions with N points, the state of the art solvers have an iterative O(N⁴) complexity when using an arbitrary loss function, making most of the real world problems intractable. In this paper, we introduce a new iterative way to approximate GW, called Sampled Gromov Wasserstein, which uses the current estimate of the transport plan to guide the sampling of cost matrices. This simple idea, supported by theoretical convergence guarantees, comes with a O(N²) solver. A special case of Sampled Gromov Wasserstein, which can be seen as the natural extension of the well known Sliced Wasserstein to distributions lying in different spaces, reduces even further the complexity to O(N log N). Our contributions are supported by experiments on synthetic and real datasets.

最优传输 (OT) 已被证明是在机器学习中比较概率分布的强大工具，但处理位于不同空间的概率度量仍然是一个悬而未决的问题。为了解决这个问题，Gromov Wasserstein 距离 (GW) 仅考虑分布内成对 (dis) 相似性。然而，对于具有N个点的两个（离散）分布，最先进的求解器在使用任意损失函数时具有迭代O ( N ⁴ ) 复杂度，这使得大多数现实世界问题变得棘手。在本文中，我们介绍了一种新的迭代方法来近似 GW，称为Sampled Gromov Wasserstein，它使用运输计划的当前估计来指导成本矩阵的抽样。这个由理论收敛保证支持的简单想法带有O ( N ² ) 求解器。Sampled Gromov Wasserstein 的一个特例，可以看作是众所周知的 Sliced Wasserstein 对不同空间分布的自然扩展，进一步将复杂性降低到O ( N log N )。我们的贡献得到了合成和真实数据集实验的支持。