Mathematics Research Institute


Weak Limits for Empirical Entropic Optimal Transport

Alberto González Sanz (Columbia University)

Fecha: 10/07/2024 12:00
Lugar: Sala de Grados I, Facultad de Ciencias

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.