Sharpening the convergence exponent and characterizing worst-case instances

Determine whether the generic-case convergence rate of the Projected Bures--Wasserstein Gradient Descent analysis admits a sharper exponent, or whether the worst-case convergence rate is attained only on rare instances.

Background

The paper establishes a dimension-independent linear convergence guarantee with iteration complexity proportional to $\kappa^{3/2}\log(1/\epsilon)$ for Projected Bures--Wasserstein Gradient Descent applied to both the Bures--Wasserstein barycenter problem and the invariant matrix projection problem. Numerical experiments show substantially faster convergence than this worst-case bound, even after refinement of the condition-number constants.

The unresolved issue is whether the observed empirical speed reflects a genuinely sharper generic-case exponent than the proven $\kappa^{3/2}$ dependence, or instead results from typical instances being far from the worst-case configurations. Resolving this would clarify the gap between the theoretical guarantee and the behavior observed in experiments.