Hit-and-Run is a coordinate-free Gibbs sampler, yet the quantitative advantages of its coordinate-free property remain largely unexplored beyond empirical studies. In this paper, we prove sharp estimates for the Wasserstein contraction of Hit-and-Run in Gaussian target measures via coupling methods and conclude mixing time bounds. Our results uncover ballistic and superdiffusive convergence rates in certain settings. Furthermore, we extend these insights to a coordinate-free variant of the randomized Kaczmarz algorithm, an iterative method for linear systems, and demonstrate analogous convergence rates. These findings offer new insights into the advantages and limitations of coordinate-free methods for both sampling and optimization.

翻译：命中-逃逸算法是一种无坐标的吉布斯采样器，但其无坐标特性的量化优势在实证研究之外仍鲜有探索。本文通过耦合方法证明了命中-逃逸算法在高斯目标测度下的Wasserstein收缩率精确估计，并推导出混合时间界。我们的结果揭示了特定场景下的弹道收敛与超扩散收敛速率。进一步地，我们将这些发现拓展至随机Kaczmarz算法（一种线性系统迭代解法）的无坐标变体，并证明了类似的收敛速率。这些研究结果为无坐标方法在采样与优化领域的优势与局限提供了新的理论洞见。