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When faced with a constant target density, geodesic slice sampling on thesphere simplifies to a geodesic random walk. We prove that this random walk isWasserstein contractive and that its contraction rate stabilizes withincreasing dimension instead of deteriorating arbitrarily far. Thisdemonstrates that the performance of geodesic slice sampling on the sphere canbe entirely robust against dimension-increases, which had not been knownbefore. Our result is also of interest due to its implications regarding thepotential for dimension-independent performance by Gibbsian polar slicesampling, which is an MCMC method on $\mathbb{R}^d$ that implicitly usesgeodesic slice sampling on the sphere within its transition mechanism.

A dimension-independent bound on the Wasserstein contraction rate of geodesic slice sampling on the sphere for uniform target