Statistical inference with regularized optimal transport.

The article develops a unified statistical framework for analyzing empirical regularized optimal transport (OT) distances, focusing on three main regularization methods: slicing, smoothing with compactly supported kernels, and entropic penalty. It establishes limit distributions, semiparametric efficiency of plug-in estimators, and bootstrap consistency for these regularized OT distances, addressing gaps in existing literature and extending results to dependent data settings. For sliced Wasserstein distances, the framework provides comprehensive limit theorems and efficiency results under compact support and moment conditions, including for average- and max-sliced variants. For smooth Wasserstein distances with compactly supported kernels, it demonstrates that these inherit structural and statistical properties of Gaussian-smoothed OT while enabling computational tractability. Finally, for entropic OT, the work derives central limit theorems, efficiency bounds, and bootstrap consistency, generalizing prior results and allowing for dependent data. The framework's flexibility suggests applicability beyond the studied examples, potentially benefiting broader classes of OT-related functionals.