We propose new Markov chain Monte Carlo algorithms to sample a uniform distribution on a convex body $K$. Our algorithms are based on the Alternating Sampling Framework/proximal sampler, which uses Gibbs sampling on an augmented distribution and assumes access to the so-called restricted Gaussian oracle (RGO). The key contribution of this work is the efficient implementation of RGO for uniform sampling on $K$ via rejection sampling and access to either a projection oracle or a separation oracle on $K$. In both oracle cases, we establish non-asymptotic complexities to obtain unbiased samples where the accuracy is measured in R\'enyi divergence or $\chi^2$-divergence.

翻译：本文提出了一种新的马尔可夫链蒙特卡洛算法，用于在凸体$K$上采样均匀分布。我们的算法基于交替采样框架/邻近采样器，该框架通过对增广分布进行吉布斯采样，并假设能够访问所谓的受限高斯预言机（RGO）。本工作的核心贡献在于，通过拒绝采样并结合对$K$的投影预言机或分离预言机的访问，实现了在$K$上进行均匀采样的RGO高效实现。针对两种预言机情况，我们建立了获得无偏样本的非渐近复杂度，其中精度以Rényi散度或$\chi^2$-散度衡量。