Improve the semi-dual Hessian lower bound and downstream constants

Determine whether more sophisticated spectral analysis can yield an improved Hessian lower-bound constant \(\kappa\) for the entropic semi-dual functional in the discrete-to-subGaussian regime, thereby improving the downstream statistical constants.

Background

The paper establishes a strong-concavity bound for the semi-dual objective through a direct lower bound on the nonzero eigenvalues of its Hessian. The resulting constant $\kappa$ has exponential dependence on the support radius, the subGaussian proxy, and the inverse regularization parameter.

The authors note that their proof is cruder than spectral approaches used in related unregularized analyses. Improving $\kappa$ would sharpen the constants appearing in the convergence rates for the empirical dual potentials, coupling density, barycentric projections, and related quantities.