Hölder continuity of the OT projection map

Prove Hölder continuity of the projection map that sends a conditional transition operator Q to the set of minimizers of cross-entropy over the entropic optimal-transport model set, and determine whether a universal Hölder exponent exists.

Background

Under misspecification, the estimator is interpreted as converging to the cross-entropy projection of the true conditional transition operator onto the model set generated by entropic optimal transport. The paper establishes convergence under compactness, a uniform law of large numbers, and uniqueness of the pseudo-true point, but it does not establish regularity of the projection as the observed operator varies.

Numerical experiments estimate a setting-dependent effective exponent, denoted α_eff, but the paper explicitly distinguishes this empirical quantity from a universal theoretical Hölder exponent. Such a continuity result would quantify robustness of the misspecified estimator to perturbations in the observed transition operator.

References

Conjectures: (i) H"older continuity of the projection map $Q\mapsto\arg\min_{Q'\in\mathcal{M}{\mathrm{OT}}}\mathrm{CE}(Q|Q')$; the $\alpha{\mathrm{eff}}$ in the experiments is only a finite-setting empirical effective exponent, not a universal theoretical exponent;

— Sinkhorn Linearization and the Spectral Proxy: Unifying the Statistical and Algorithmic Theory of Feature-Parameterized Inverse Optimal Transport via a Single Spectral Sandwich  (2608.13201 - Dong et al., 13 Aug 2026) in Section 1, subsection “Open problems” (also discussed in Remark 2.14 and Section 6, O5)