The Monge Gap: A Regularizer to Learn All Transport Maps

Optimal transport (OT) theory has 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 $\mathcal{P}(\mathbb{R}^d)$ into another must be the gradient of a convex function. To exploit that result, Makkuva et. al (2020); Korotin et. al (2020) consider maps $T=\nabla f_\theta$, where $f_\theta$ is an input convex neural network (ICNN), as defined by Amos et. al (2017), and fit $\theta$ with SGD using samples. Despite their mathematical elegance, fitting OT maps with ICNNs raises many challenges, due notably to the many constraints imposed on $\theta$; the need to approximate the conjugate of $f_\theta$; or the limitation that they only work for the squared-Euclidean cost. More generally, we question the relevance of using Brenier's result, which only applies to densities, to constrain the architecture of candidate maps fitted on samples. Motivated by these limitations, we propose a radically different approach to estimating OT maps: Given a cost $c$ and a reference measure $\rho$, we introduce a regularizer, the Monge gap $\mathcal{M}^c_{\rho}(T)$ of a map $T$. That gap quantifies how far a map $T$ deviates from the ideal properties we expect from a $c$-OT map. In practice, we drop all architecture requirements for $T$ and simply minimize a distance (e.g., the Sinkhorn divergence) between $T\sharp\mu$ and $\nu$, regularized by $\mathcal{M}^c_\rho(T)$. We study $\mathcal{M}^c_{\rho}$ and show how our simple pipeline significantly outperforms other baselines in practice.