In this paper, we present a novel and principled approach to learn the
optimal transport between two distributions, from samples. Guided by the
optimal transport theory, we learn the optimal Kantorovich potential which
induces the optimal transport map. This involves learning two convex functions,
by solving a novel minimax optimization. Building upon recent advances in the
field of input convex neural networks, we propose a new framework where the
gradient of one convex function represents the optimal transport mapping.
Numerical experiments confirm that we learn the optimal transport mapping. This
approach ensures that the transport mapping we find is optimal independent of
how we initialize the neural networks. Further, target distributions from a
discontinuous support can be easily captured, as gradient of a convex function
naturally models a {\em discontinuous} transport mapping.

本文提出了一种新的、原则性的方法来从样本中学习两个分布之间的最优传输，学习方法基于最优传输理论并涉及解决一个新的极小极大优化问题，通过最优 Kantorovich 势量级诱导最优传输映射，借鉴最近在输入凸型神经网络领域的进展，提出了一个新的框架，其中一个凸函数的梯度表示最优传输映射。数值实验表明，我们学习到了最优传输映射，这一方法确保我们发现的传输映射独立于神经网络的初始化方式。而且，由于凸函数的梯度自然地模拟了不连续的传输映射，因此可以轻松捕捉具有不连续支持的目标分布。

通过输入凸神经网络实现最优传输映射

Optimal transport mapping via input convex neural networks

In this paper, we introduce methods from convex optimization to solve the
multimarginal transport type problems arise in the context of density
functional theory. Convex relaxations are used to provide outer approximation
to the set of $N$-representable 2-marginals and 3-marginals, which in turn
provide lower bounds to the energy. We further propose rounding schemes to
obtain upper bound to the energy. Numerical experiments demonstrate a gap of
the order of $10^{-3}$ to $10^{-2}$ between the upper and lower bounds. The
Kantorovich potential of the multi-marginal transport problem is also
approximated with a similar accuracy.

本文应用凸优化理论解决密度泛函理论中出现的多重边际输运类型问题，采用凸松弛方法提供 $N$- 可表示的二重与三重边际的外部近似来提供能量下界，并进一步提出取整方案以获得能量上界，数值实验结果表明上下界之间的误差在 $10^{-3}$ 到 $10^{-2}$ 的数量级，同时还对多重边际输运问题中的 Kantorovich 势进行了类似精度的近似。

严格相关密度泛函理论的凸松弛方法

Convex relaxation approaches for strictly correlated density functional  theory

In this work, we show the intrinsic relations between optimal transportation
and convex geometry, especially the variational approach to solve Alexandrov
problem: constructing a convex polytope with prescribed face normals and
volumes. This leads to a geometric interpretation to generative models, and
leads to a novel framework for generative models. By using the optimal
transportation view of GAN model, we show that the discriminator computes the
Kantorovich potential, the generator calculates the transportation map. For a
large class of transportation costs, the Kantorovich potential can give the
optimal transportation map by a close-form formula. Therefore, it is sufficient
to solely optimize the discriminator. This shows the adversarial competition
can be avoided, and the computational architecture can be simplified.
Preliminary experimental results show the geometric method outperforms WGAN for
approximating probability measures with multiple clusters in low dimensional
space.

本文提出一种基于凸几何和最优输运的可变分模型生成方法，通过对 GAN 模型的最优输运视角，表明判别器计算 Kantorovich 势，生成器计算输运映射