  Optimal transport (OT) is a versatile framework for comparing probability
measures, with many applications to statistics, machine learning, and applied
mathematics. However, OT distances suffer from computational and statistical
scalability issues to high dimensions, which motivated the study of regularized
OT methods like slicing, smoothing, and entropic penalty. This work establishes
a unified framework for deriving limit distributions of empirical regularized
OT distances, semiparametric efficiency of the plug-in empirical estimator, and
bootstrap consistency. We apply the unified framework to provide a
comprehensive statistical treatment of: (i) average- and max-sliced
$p$-Wasserstein distances, for which several gaps in existing literature are
closed; (ii) smooth distances with compactly supported kernels, the analysis of
which is motivated by computational considerations; and (iii) entropic OT, for
which our method generalizes existing limit distribution results and
establishes, for the first time, efficiency and bootstrap consistency. While
our focus is on these three regularized OT distances as applications, the
flexibility of the proposed framework renders it applicable to broad classes of
functionals beyond these examples.
