Neural optimal transport techniques mostly use Euclidean cost functions, such
as $\ell^1$ or $\ell^2$. These costs are suitable for translation tasks between
related domains, but they are hardly applicable to problems where a specific
non-Euclidean optimality of the mapping is required such as dataset transfer.
To tackle this issue, we introduce a novel neural network-based algorithm to
compute optimal transport plans and maps for general cost functionals. Such
functionals provide more flexibility and allow using auxiliary information,
such as class labels, to construct the required transport map. Our method is
based on a saddle point reformulation of the optimal transport problem and
generalizes prior methods for weak and strong transport cost functionals. As an
application, we construct a functional to map data distributions with
preserving the class-wise structure of data.

本文介绍了一种基于神经网络的新算法，用于计算一般的成本功能的最优输运方案和映射，该算法通过鞍点重构最优输运问题并推广到先前在弱和强输运成本功能中的方法，最终构建了一个可保存数据类别结构的数据分布映射功能。

具有通用代价函数的神经最优输运

Neural Optimal Transport with General Cost Functionals

We explore the question of which shape a manifold is compelled to take when
immersed in another one, provided it must be the extremum of some functional.
We consider a family of functionals which depend quadratically on the extrinsic
curvatures and on projections of the ambient curvatures. These functionals
capture a number of physical setups ranging from holography to the study of
membranes and elastica. We present a detailed derivation of the equations of
motion, known as the shape equations, placing particular emphasis on the issue
of gauge freedom in the choice of normal frame. We apply these equations to the
particular case of holographic entanglement entropy for higher curvature three
dimensional gravity and find new classes of entangling curves. In particular,
we discuss the case of New Massive Gravity where we show that non-geodesic
entangling curves have always a smaller on-shell value of the entropy
functional. Then we apply this formalism to the computation of the entanglement
entropy for dual logarithmic CFTs. Nevertheless, the correct value for the
entanglement entropy is provided by geodesics. Then, we discuss the importance
of these equations in the context of classical elastica and comment on terms
that break gauge invariance.

探讨曲面嵌入另一个曲面时在满足极大限制的情况下可能采取的形状，提供了一种描述物理设备的函数族，应用方程式到全息学和膜和弹性的研究中，对测量量子纠缠熵的重力模型进行了详细推导，并描述了正确的测量纠缠熵值得方式