Hug is a recently proposed iterative mapping used to design efficient updates in Markov chain Monte Carlo (MCMC) methods. Hug generates proposals that remain very close to hypersurfaces (level sets) of constant probabilty density. We analyse a generalization of Hug from hypersurfaces to manifolds of arbitrary dimensions, not necessarily arising in a sampling context. The analysis is based on interpreting, in a nonstandard way, Hug as a consistent discretization of a system of differential equations with a rather complicated structure. The proof of convergence of this discretization includes a number of unusual features we explore fully, in particular a supraconvergence property is established, whereby second order of convergence is attained with consistency of the first order. We uncover and discuss an unexpected property of the solutions of the underlying dynamical system that manifest itself by the existence of Hug trajectories that fail to cover the manifold of interest.

翻译：Hug是近期提出的一种迭代映射，用于设计马尔可夫链蒙特卡洛（MCMC）方法中的高效更新。Hug生成的提案始终非常接近概率密度恒定的超曲面（水平集）。我们将Hug从超曲面推广到任意维度的流形进行分析，该分析不一定在采样背景下进行。此分析基于以一种非标准方式将Hug解释为一类结构相当复杂的微分方程系统的相容离散化。该离散化收敛性的证明包含若干我们深入探究的非寻常特性，特别是建立了超收敛性质——即仅具备一阶相容性时却达到了二阶收敛精度。我们发现并讨论了底层动力系统解的一个意外性质，该性质表现为存在无法覆盖目标流形的Hug轨迹。