Sample complexity of unbalanced entropic OT

Optimal transport (OT) is a powerful tool for comparing probability measures, but standard balanced OT is often too rigid for real-world data where mass may be created or destroyed. While entropic regularization makes OT computationally efficient, statistical guarantees for the resulting transport plans in the unbalanced setting remain underdeveloped. This paper addresses this gap by deriving finite-sample complexity bounds for the optimal coupling in unbalanced entropic OT.

The authors analyze the dual formulation of unbalanced entropic OT. A primary challenge is that the dual objective lacks the translation (gauge) invariance found in balanced OT, which complicates stability analysis. The authors introduce a translation-invariant envelope functional that effectively quotients out the scalar translation direction. They prove that this envelope is strongly convex and admits a compact minimization domain, providing a stable geometric framework for the dual problem. By leveraging these properties, they derive high-probability bounds on the deviation between empirical and population-level transport plans.

The paper demonstrates that the translation-invariant dual formulation is well-behaved under mild assumptions. Specifically, the authors show that the dual potentials are uniformly bounded and that the envelope functional is strongly convex with respect to the intrinsic degrees of freedom. This geometric stability allows for the derivation of explicit finite-sample convergence rates for the transport plan. These results confirm that entropic regularization not only enables scalable computation via Sinkhorn-type solvers but also improves the statistical stability of the estimator, effectively mitigating the curse of dimensionality.