Title: OTCOP: Learning optimal transport maps via constraint optimizations. Abstract: The approximation power of neural networks makes it an ideal tool to learn optimal transport maps. However, existing methods are mostly based on the Kantorovich duality and require regularization and/or special network structures such as Input Convex Neural Networks (ICNN). In this paper, we propose a direct constraint optimization algorithm for the computation of optimal transport maps based on the  Monge formulation. We solve this constraint optimization problem by using three different methods: the penalty method, the augmented Lagrangian method, and the alternating direction method of multipliers method (AMDD). We demonstrate a significant improvement in the accuracy of learned optimal transport maps on benchmarks. Moreover, we show that our methods reduce the regularization effects and accurately learn the target distributions at lower transport cost.
