On amortizing convex conjugates for optimal transport
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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
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