  Entropic optimal transport (EOT) presents an effective and computationally
viable alternative to unregularized optimal transport (OT), offering diverse
applications for large-scale data analysis. In this work, we derive novel
statistical bounds for empirical plug-in estimators of the EOT cost and show
that their statistical performance in the entropy regularization parameter
$\epsilon$ and the sample size $n$ only depends on the simpler of the two
probability measures. For instance, under sufficiently smooth costs this yields
the parametric rate $n^{-1/2}$ with factor $\epsilon^{-d/2}$, where $d$ is the
minimum dimension of the two population measures. This confirms that empirical
EOT also adheres to the lower complexity adaptation principle, a hallmark
feature only recently identified for unregularized OT. As a consequence of our
theory, we show that the empirical entropic Gromov-Wasserstein distance and its
unregularized version for measures on Euclidean spaces also obey this
principle. Additionally, we comment on computational aspects and complement our
findings with Monte Carlo simulations. Our techniques employ empirical process
theory and rely on a dual formulation of EOT over a single function class.
Crucial to our analysis is the observation that the entropic
cost-transformation of a function class does not increase its uniform metric
entropy by much.
