Entropic optimal transport bias asymptotics with boundary effects

Establish an entropic analogue of the quadratically regularized transport-map bias asymptotics for marginal supports with boundaries, including an additional boundary term of order √ε and the resulting L^p approximation rate ε^{(1+1/p)/2} for the entropic conditional mean.

Background

The paper derives sharp global Lp asymptotics for two approximations to the Brenier map under quadratic regularization: the gradient of the dual potential and the conditional mean. The dominant contribution arises from a boundary layer of width ℓ=ε{1/(d+2)} and amplitude ℓ.

For entropic regularization, the authors conjecture that analogous boundary phenomena should occur when the marginal supports have boundaries, and more generally that the bias expansion for the entropic conditional mean can be established. The conjecture specifies that the expansion should contain a boundary term of order √ε and that this boundary layer should slow the approximation rate to ε{(1+1/p)/2} in Lp.

References

We conjecture that similar phenomena arise in the entropic case when the marginal supports have a boundary, and more generally that an entropic analogue of \cref{thm:main-bias} can be shown. The expansion would then carry an additional boundary term of order $\sqrt{}$, and the approximation would slow to $|m_-T|_{Lp(\mu)}\asymp {(1+1/p)/2}$.

— 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)