Recently, the Gromov-Wasserstein Optimal Transport (GWOT) problem has attracted the special attention of the ML community. In this problem, given two distributions supported on two (possibly different) spaces, one has to find the most isometric map between them. In the discrete variant of GWOT, the task is to learn an assignment between given discrete sets of points. In the more advanced continuous formulation, one aims at recovering a parametric mapping between unknown continuous distributions based on i.i.d. samples derived from them. The clear geometrical intuition behind the GWOT makes it a natural choice for several practical use cases, giving rise to a number of proposed solvers. Some of them claim to solve the continuous version of the problem. At the same time, GWOT is notoriously hard, both theoretically and numerically. Moreover, all existing continuous GWOT solvers still heavily rely on discrete techniques. Natural questions arise: to what extent existing methods unravel GWOT problem, what difficulties they encounter, and under which conditions they are successful. Our benchmark paper is an attempt to answer these questions. We specifically focus on the continuous GWOT as the most interesting and debatable setup. We crash-test existing continuous GWOT approaches on different scenarios, carefully record and analyze the obtained results, and identify issues. Our findings experimentally testify that the scientific community is still missing a reliable continuous GWOT solver, which necessitates further research efforts. As the first step in this direction, we propose a new continuous GWOT method which does not rely on discrete techniques and partially solves some of the problems of the competitors. Our code is available at https://github.com/Ark-130994/GW-Solvers.

翻译：近年来，Gromov-Wasserstein最优传输问题引起了机器学习领域的特别关注。在该问题中，给定两个支撑在（可能不同的）空间上的分布，需要找到它们之间最具等距性的映射。在离散版本的GWOT中，任务是在给定的离散点集之间学习一种对应关系。而在更高级的连续形式中，目标是根据从未知连续分布中独立同分布抽取的样本，恢复这些分布之间的参数化映射。GWOT背后清晰的几何直觉使其成为多个实际应用场景的自然选择，从而催生了若干求解器的提出。其中部分方法声称能够解决该问题的连续版本。与此同时，GWOT在理论和数值计算上都 notoriously 困难。此外，所有现有的连续GWOT求解器仍然严重依赖离散技术。由此产生了一些自然问题：现有方法在多大程度上揭示了GWOT问题、它们遇到了哪些困难、以及在何种条件下能够成功。我们的基准论文试图回答这些问题。我们特别关注连续GWOT这一最具争议且引人关注的设定。我们在不同场景下对现有连续GWOT方法进行压力测试，仔细记录并分析所得结果，识别其中存在的问题。我们的发现通过实验证明，科学界目前仍缺乏可靠的连续GWOT求解器，这需要进一步的研究努力。作为该方向的第一步，我们提出了一种不依赖离散技术的新连续GWOT方法，该方法部分解决了现有竞争方法的一些问题。我们的代码公开于https://github.com/Ark-130994/GW-Solvers。