On amortizing convex conjugates for optimal transport

Brandon Amos

Keywords: optimal transport, wasserstein-2, convex conjugate, c-transform, amortized optimization

TL;DR: State-of-the art continuous Wasserstein-2 potential learning, and along the way I improved Jax's L-BFGS implementation to run in 3% of the time for solving batches of optimization problems

Abstract: 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.

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