We present a new random walk for uniformly sampling high-dimensional convex bodies. It achieves state-of-the-art runtime complexity with stronger guarantees on the output than previously known, namely in R\'enyi divergence (which implies TV, $\mathcal{W}_2$, KL, $\chi^2$). The proof departs from known approaches for polytime algorithms for the problem -- we utilize a stochastic diffusion perspective to show contraction to the target distribution with the rate of convergence determined by functional isoperimetric constants of the stationary density.

翻译：我们提出了一种新的随机游走方法，用于对高维凸体进行均匀采样。该方法在输出上比先前已知方法具有更强的保证（即在Rényi散度下，这涵盖了总变差距离、$\mathcal{W}_2$距离、KL散度、$\chi^2$散度），同时实现了最先进的运行时复杂度。其证明脱离了该问题多项式时间算法的已知方法——我们利用随机扩散视角，展示了目标分布的收缩性，其收敛速率由平稳密度的函数等周常数决定。