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.
翻译:这项工作涉及潜在和混合的正正统优化运输潜力和组合的无线分布, 以美元计算。 我们首先提供辛克霍恩潜在潜力的核心限值 -- -- 解决双重问题的方案 -- -- 以美元计高山进程。 然后我们获得组合的微弱限度 -- -- 原始问题的解决方案 -- -- 以不可磨灭的功能来评估, 证明了对高比亚进程的推测值{ChaosDecom}。 在这两种情况下, 其限值是真实的高比随机变量。 最后, 我们考虑在两种假设下, 美元=0: = 美元= ⁇ 美元 或 美元= = 1 = = 美元 = = 美元 = = 1 = = 美元 = = = ⁇ 或 美元= = = = = = 美元 = 。 低于H_ 0 的限值是高比亚进程的假设值, 在 $_ 1 美元 下, 其限值是真实的随机变量 。 我们还提供了一个以最高比例 。