Optimal transport theory has provided machine learning with several tools to infer a push-forward map between densities from samples. While this theory has recently seen tremendous methodological developments in machine learning, its practical implementation remains notoriously difficult, because it is plagued by both computational and statistical challenges. Because of such difficulties, existing approaches rarely depart from the default choice of estimating such maps with the simple squared-Euclidean distance as the ground cost, $c(x,y)=\|x-y\|^2_2$. We follow a different path in this work, with the motivation of \emph{learning} a suitable cost structure to encourage maps to transport points along engineered features. We extend the recently proposed Monge-Bregman-Occam pipeline~\citep{cuturi2023monge}, that rests on an alternative cost formulation that is also cost-invariant $c(x,y)=h(x-y)$, but which adopts a more general form as $h=\tfrac12 \ell_2^2+\tau$, where $\tau$ is an appropriately chosen regularizer. We first propose a method that builds upon proximal gradient descent to generate ground truth transports for such structured costs, using the notion of $h$-transforms and $h$-concave potentials. We show more generally that such a method can be extended to compute $h$-transforms for entropic potentials. We study a regularizer that promotes transport displacements in low-dimensional spaces, and propose to learn such a basis change using Riemannian gradient descent on the Stiefel manifold. We show that these changes lead to estimators that are more robust and easier to interpret.

本文提出了一种基于推向前映射和学习适当代价结构的方法，通过使用 Monge-Bregman-Occam 管线，使用 $h$-变换和 $h$-凹潜力生成适应结构化代价的基本真实传输，并提出一种学习低维空间中传输位移的正则化方法，通过 Riemannian 梯度下降对 Stiefel 流形进行基础变化，从而得到更加稳健和易于解释的估计量。

基于蒙特序列的结构性位移学习成本