标题： 超越乘积参考耦合的熵最优传输：欧几里得空间上的高斯情形 显示英文标题

标题： Entropic optimal transport beyond product reference couplings: the Gaussian case on Euclidean space

摘要： 带平方欧几里得代价的最优传输问题在于找到两个输入测度之间的耦合，以最大化相关性。 因此，最优耦合通常相对于勒贝格测度是奇异的。 用熵项正则化最优传输问题会产生一个称为熵最优传输的近似方法。 熵惩罚会引导产生的耦合趋向于具有所需特性的参考测度。 例如，当寻求扩散耦合时，最常用的参考测度是勒贝格测度和两个输入测度的乘积。 在本工作中，我们研究参考耦合不一定假设为乘积的情况。 我们以高斯情况作为激励范例，并将这种更一般的最优传输准则简化为一个矩阵优化问题。 这种简化使我们能够从原始变量和对偶变量的角度完整地描述解。 我们认为，在统计背景下，参考测度的灵活性可能很重要，例如当有先验信息时，当对要耦合的测度存在不确定性时，或者当使用熵问题来估计未正则化的传输问题时减少偏差。 特别是，我们在数值例子中展示了选择合适的参考计划可以减少由熵惩罚引起的偏差。 显示英文摘要

摘要： The optimal transport problem with squared Euclidean cost consists in finding a coupling between two input measures that maximizes correlation. Consequently, the optimal coupling is often singular with respect to Lebesgue measure. Regularizing the optimal transport problem with an entropy term yields an approximation called entropic optimal transport. Entropic penalties steer the induced coupling toward a reference measure with desired properties. For instance, when seeking a diffuse coupling, the most popular reference measures are the Lebesgue measure and the product of the two input measures. In this work, we study the case where the reference coupling is not necessarily assumed to be a product. We focus on the Gaussian case as a motivating paradigm, and provide a reduction of this more general optimal transport criterion to a matrix optimization problem. This reduction enables us to provide a complete description of the solution, both in terms of the primal variable and the dual variables. We argue that flexibility in terms of the reference measure can be important in statistical contexts, for instance when one has prior information, when there is uncertainty regarding the measures to be coupled, or to reduce bias when the entropic problem is used to estimate the un-regularized transport problem. In particular, we show in numerical examples that choosing a suitable reference plan allows to reduce the bias caused by the entropic penalty.