The Monge Gap: A Regularizer to Learn All Transport Maps

Optimal transport (OT) theory has been been used in machine learning to study and characterize maps that can push-forward efficiently a probability measure onto another.
Recent works have drawn inspiration from Brenier’s theorem, which states that when the ground cost is the squared-Euclidean distance, the “best” map to morph a continuous measure in into another must be the gradient of a convex function.
To exploit that result, , Makkuva et al. (2020); Korotin et al. (2020) consider maps , where is an input convex neural network (ICNN), as defined by Amos et al. 2017, and fit with SGD using…

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