  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.
