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 Rényi divergence (which implies TV, $\mathcal{W}_2$, KL, $χ^2$). 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 target distribution.

翻译：我们提出一种用于均匀采样高维凸体的新随机游走算法。该算法实现了当前最优的运行时复杂度，并在输出质量上提供了比以往方法更强的保证，具体体现在Rényi散度（该散度蕴含总变差距离、$\mathcal{W}_2$距离、KL散度、$χ^2$散度）中。其证明突破了该问题现有多项式时间算法的传统思路——我们采用随机扩散视角，证明算法向目标分布的收缩过程，其收敛速率由目标分布的函数等周常数决定。