Sample complexity for entropic optimal transport with radial cost
Ruiyu Han, Johannes Wiesel
TL;DR
The paper derives sharp (up to logarithms) sample-complexity bounds for entropy-regularized OT with radial costs on $\mathbb{R}^d$ under exponential tail decay. By truncating to growing balls and exploiting a three-term decomposition, the authors bound the empirical EOT error with terms reflecting tail probabilities, intrinsic dimension via covering numbers, and the Lipschitz structure of radial costs. The main result shows a $1/\sqrt{n}$ rate modulated by logarithmic factors and the generalized covering-number terms, applicable to subgaussian/subexponential marginals and costs $c(x,y)=|x-y|^p$, $p\ge 2$. This advances understanding of EOT in unbounded settings and highlights the role of intrinsic-dimension measures in high-dimensional statistical OT. The analysis yields practical corollaries for compact supports, manifolds, and semi-discrete settings, with explicit dependence on tails, moments, and dimension.
Abstract
We prove a new sample complexity result for entropy regularized optimal transport. Our bound holds for probability measures on $\mathbb R^d$ with exponential tail decay and for radial cost functions that satisfy a local Lipschitz condition. It is sharp up to logarithmic factors, and captures the intrinsic dimension of the marginal distributions through a generalized covering number of their supports. Examples that fit into our framework include subexponential and subgaussian distributions and radial cost functions $c(x,y)=|x-y|^p$ for $p\ge 2.$