Convergence of transport plans and potentials under divergence regularization

Investigate the convergence properties of optimal transport plans and optimal potentials in divergence-regularized optimal transport, providing a separate and detailed asymptotic analysis beyond the transport cost considered in this work.

Background

While this paper establishes sample complexity and central limit theorems for the divergence-regularized transport cost, it does not analyze the asymptotic behavior of other key objects such as optimal transport plans and potentials.

The authors explicitly defer this analysis, indicating a concrete open direction focused on the convergence properties of these objects under divergence regularization.

References

Second, while this paper focuses specifically on the convergence of the transport cost, other central objects in optimal transport—such as the transport plans and the optimal potentials—warrant a separate and detailed asymptotic analysis. We leave the investigation of their convergence properties for future work.

— General Divergence Regularized Optimal Transport: Sample Complexity and Central Limit Theorems  (2510.02489 - Yang et al., 2 Oct 2025) in Section 5 (Conclusion)

We conjecture that an entropic analogue of \cref{thm:corrector} holds, with potential corrections of order $$ after subtracting the common term $\frac{d}{4}\log(2\pi)$ and a boundary layer of width $\sqrt{}$.

— Geometry and Convergence of Quadratically Regularized Optimal Transport II  (2610.02077 - González-Sanz et al., 1 Oct 2026) in Section 1, subsection “Related literature” (discussion of entropic regularization)