论文标题
熵正则最佳运输的较弱限制；潜力，计划和分歧
Weak limits of entropy regularized Optimal Transport; potentials, plans and divergences
论文作者
论文摘要
这项工作涉及熵正规化最佳运输电位和耦合的渐近分布，以$ \ r^d $中的紧凑型概率。我们首先提供了Sinkhorn电位的中心限制定理 - 双重问题的解决方案 - 作为$ \ cs $的高斯过程。然后，我们获得了对整合函数的评估耦合的弱极限 - 原始问题的解决方案，证明了\ cite {chaosdecom}的猜想。在这两种情况下，它们的极限都是一个真正的高斯随机变量。最后，我们考虑两个假设下的熵sindhorn脱落的弱极限$ h_0：\ {\ rm p} = {\ rm q} $或$ h_1：\ {\ rm p} \ neq {\ neq {\ rm q} $。在$ h_0 $下，限制是一种二次形式，在sobolev空间中应用于高斯流程，而$ h_1 $以下，限制为高斯。我们还提供了$ h_0 $以下限制的不同表征，就i.i.d的无限总和而言。标准高斯随机变量的序列。这样的结果可以基于熵正规化的最佳运输来实现统计推断。
This work deals with the asymptotic distribution of both potentials and couplings of entropic regularized optimal transport for compactly supported probabilities in $\R^d$. We first provide the central limit theorem of the Sinkhorn potentials -- the solutions of the dual problem -- as a Gaussian process in $\Cs$. Then we obtain the weak limits of the couplings -- the solutions of the primal problem -- evaluated on integrable functions, proving a conjecture of \cite{ChaosDecom}. In both cases, their limit is a real Gaussian random variable. Finally we consider the weak limit of the entropic Sinkhorn divergence under both assumptions $H_0:\ {\rm P}={\rm Q}$ or $H_1:\ {\rm P}\neq{\rm Q}$. Under $H_0$ the limit is a quadratic form applied to a Gaussian process in a Sobolev space, while under $H_1$, the limit is Gaussian. We provide also a different characterisation of the limit under $H_0$ in terms of an infinite sum of an i.i.d. sequence of standard Gaussian random variables. Such results enable statistical inference based on entropic regularized optimal transport.
