Determine whether slower fully subGaussian rates are intrinsic

Determine whether the \(n^{-1/2}\) squared-error rates known for entropic optimal transport in fully subGaussian settings are intrinsic to those settings or instead result from limitations of existing statistical analyses.

Background

The paper proves parametric $n^{-1}$ squared-error rates for several entropic optimal transport objects when one marginal is finitely supported and the other is subGaussian. It contrasts these results with slower $n^{-1/2}$ rates reported for fully subGaussian settings.

The authors leave unresolved whether the slower fully subGaussian rates reflect a fundamental statistical barrier or merely the weakness of current proof techniques. Resolving this issue would clarify the scope of lower-complexity adaptation in entropic optimal transport.