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. To overcome this, the computation of the conjugate can be approximated with amortized optimization, which learns a model to predict the conjugate. I show that combining amortized approximations to the conjugate with a solver for fine-tuning significantly improves the quality of transport maps learned for the Wasserstein-2 benchmark by Korotin et al. (2021a) 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：本文聚焦于计算欧几里得Wasserstein-2最优输运问题中出现的凸共轭运算。该共轭运算（也称为Legendre-Fenchel共轭或c-变换）被认为难以计算，且在实践中，Wasserstein-2方法因无法在连续空间中对偶势能进行精确共轭而受到限制。为解决此问题，可通过摊销优化逼近共轭运算的计算——即训练一个模型来预测共轭值。研究证明，将共轭的摊销近似与精调求解器相结合，能显著提升Korotin等人(2021a)提出的Wasserstein-2基准测试中输运图的质量，并可建模文献中诸多二维耦合与流形。本文中所有基线、方法与求解器均公开于http://github.com/facebookresearch/w2ot。