# The Monge Gap: A Regularizer to Learn All Transport Maps

AuthorsThéo Uscidda, Marco Cuturi

content type paperpublished June 2023

AuthorsThéo Uscidda, Marco Cuturi

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 $\mathcal{P}(\mathbb{R}^d)$ 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 outperforms significantly other baselines in practice.

Optimal transport (OT) theory focuses, among all maps T:Rd→RdT:\mathbb{R}^d\rightarrow \mathbb{R}^dT:Rd→Rd that can morph a probability measure onto another, on those that are the "thriftiest", i.e. such that the averaged cost c(x,T(x))c(\mathbf{x}, T(\mathbf{x}))c(x,T(x)) between x\mathbf{x}x and its image T(x)T(\mathbf{x})T(x) be as small as possible. Many computational approaches have been proposed to estimate such Monge maps when ccc is the…

See paper detailsOptimal transport (OT) theory describes general principles to define and select, among many possible choices, the most efficient way to map a probability measure onto another. That theory has been mostly used to estimate, given a pair of source and target probability measures (μ,ν)(\mu,\nu)(μ,ν), a parameterized map TθT_\thetaTθ that can efficiently map μ\muμ onto ν\nuν. In many applications, such as predicting cell responses to treatments, the…

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