When faced with a constant target density, geodesic slice sampling on the sphere simplifies to a geodesic random walk. We prove that this random walk is Wasserstein contractive and that its contraction rate stabilizes with increasing dimension instead of deteriorating arbitrarily far. This demonstrates that the performance of geodesic slice sampling on the sphere can be entirely robust against dimension-increases, which had not been known before. Our result is also of interest due to its implications regarding the potential for dimension-independent performance by Gibbsian polar slice sampling, which is an MCMC method on $\mathbb{R}^d$ that implicitly uses geodesic slice sampling on the sphere within its transition mechanism.

翻译：当面对恒定目标密度时，球面上的测地切片抽样简化为测地随机游走。我们证明该随机游走具有Wasserstein收缩性，且其收缩率随维数增加而稳定，而非无限恶化。这表明球面上测地切片抽样的性能对维数增加具有完全鲁棒性，这一性质此前尚未被认知。该结果的重要性还在于它对吉布斯极切片抽样（一种隐式利用球面测地切片抽样作为转移机制的$\mathbb{R}^d$上的MCMC方法）实现维数无关性能的潜力具有启示意义。