  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 $\mathcal{P}(\Rd)$ into another
must be the gradient of a convex function. To exploit that result, [Makkuva+
2020, Korotin+2020] consider maps $T=\nabla f_\theta$, where $f_\theta$ is an
input convex neural network (ICNN), as defined by Amos+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.
