Abstract:
In this talk, we will delve into the asymptotic distribution of potentials and couplings in entropic regularized optimal transport for compactly supported probabilities in $\mathbb{R}^d$. Our analysis begins with the central limit theorem for Sinkhorn potentials—the solutions to the dual problem—demonstrating their convergence to a Gaussian process in $\mathcal{C}(S)$. Next, we establish the weak limits of the couplings, the solutions to the primal problem, when evaluated on integrable functions. This result verifies a conjecture proposed by Harchaoui, Liu, and Pal in 2020. Finally, we consider the weak limit of the entropic Sinkhorn divergence between two distributions $P$ and $Q$ under two hypotheses: $H_0:\ {\rm P}={\rm Q}$ and $H_1:\ {\rm P}\neq{\rm Q}$. Under $H_0$, the limit is a weighted sum of independent chi-squared random variables with one degree of freedom, while under $H_1$, the limit is Gaussian. These findings provide a foundation for statistical inference based on entropic regularized optimal transport.