Optimal transport mapping via input convex neural networks
This work addresses the challenge of computing optimal transport in machine learning, offering a principled method that is robust to initialization and can model discontinuous mappings, though it appears incremental in building upon existing neural network techniques.
The paper tackles the problem of learning optimal transport mappings between two distributions from samples by proposing a novel approach that learns the optimal Kantorovich potential using input convex neural networks, with numerical experiments confirming the method's ability to learn optimal mappings and handle discontinuous supports.
In this paper, we present a novel and principled approach to learn the optimal transport between two distributions, from samples. Guided by the optimal transport theory, we learn the optimal Kantorovich potential which induces the optimal transport map. This involves learning two convex functions, by solving a novel minimax optimization. Building upon recent advances in the field of input convex neural networks, we propose a new framework where the gradient of one convex function represents the optimal transport mapping. Numerical experiments confirm that we learn the optimal transport mapping. This approach ensures that the transport mapping we find is optimal independent of how we initialize the neural networks. Further, target distributions from a discontinuous support can be easily captured, as gradient of a convex function naturally models a {\em discontinuous} transport mapping.