We analyze Riemannian Hamiltonian Monte Carlo (RHMC) for sampling a polytope defined by $m$ inequalities in $\R^n$ endowed with the metric defined by the Hessian of a self-concordant convex barrier function. We use a hybrid of the $p$-Lewis weight barrier and the standard logarithmic barrier and prove that the mixing rate is bounded by $\tilde O(m^{1/3}n^{4/3})$, improving on the previous best bound of $\tilde O(mn^{2/3})$, based on the log barrier. Our analysis overcomes several technical challenges to establish this result, in the process deriving smoothness bounds on Hamiltonian curves and extending self-concordance notions to the infinity norm; both properties appear to be of independent interest.

翻译：我们分析了黎曼哈密顿蒙特卡洛（RHMC）方法，用于对由$\R^n$中$m$个不等式定义的多面体进行采样，该多面体配备由自和谐凸障碍函数的Hessian矩阵定义的度量。我们采用$p$-刘易斯权重障碍与标准对数障碍的混合方法，并证明混合速率受$\tilde O(m^{1/3}n^{4/3})$限制，这优于先前基于对数障碍的最优边界$\tilde O(mn^{2/3})$。我们的分析克服了多项技术难题以建立该结果，在此过程中推导了哈密顿曲线的光滑性边界，并将自和谐概念扩展到无穷范数；这两个性质似乎具有独立的研究价值。