This paper focuses on computing the convex conjugate operation that arises when solving Euclidean Wasserstein-2 optimal transport problems. This conjugation, which is also referred to as the Legendre-Fenchel conjugate or $c$-transform, is considered difficult to compute and in practice, Wasserstein-2 methods are limited by not being able to exactly conjugate the dual potentials in continuous space. I show that combining amortized approximations to the conjugate with a solver for fine-tuning is computationally easy. This combination significantly improves the quality of transport maps learned for the Wasserstein-2 benchmark by Korotin et al. (2021) and is able to model many 2-dimensional couplings and flows considered in the literature. All of the baselines, methods, and solvers in this paper are available at http://github.com/facebookresearch/w2ot

翻译：本文侧重于计算在解决欧克利底安·瓦西斯坦-2最佳运输问题时产生的共振共振操作。这种共振(也称为Tulturre-Fenchel conjuge 或$c$-transform)被认为难以计算,而且在实践中,瓦西斯坦-2方法由于无法精确地将连续空间的双重潜力混为一体而受到限制。我表明,将分解近似和精细调整的解析器相结合,在计算上是容易的。这种结合极大地提高了Korotin等人(2021年)为瓦西斯坦-2基准所学的运输地图的质量,并且能够模拟文献中考虑的许多二维联动和流动。本文中的所有基线、方法和解析器都可在http://github.com/facebourseresearch/w2ot查阅。