We propose and analyze a new Markov Chain Monte Carlo algorithm that generates a uniform sample over full and non-full dimensional polytopes. This algorithm, termed "Matrix Hit and Run" (MHAR), is a modification of the Hit and Run framework. For the regime $n^{1+\frac{1}{3}} \ll m$, MHAR has a lower asymptotic cost per sample in terms of soft-O notation ($\SO$) than do existing sampling algorithms after a \textit{warm start}. MHAR is designed to take advantage of matrix multiplication routines that require less computational and memory resources. Our tests show this implementation to be substantially faster than the \textit{hitandrun} R package, especially for higher dimensions. Finally, we provide a python library based on Pytorch and a Colab notebook with the implementation ready for deployment in architectures with GPU or just CPU.

中文翻译：

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用于采样多边体的新型矩阵即插即用及其GPU实现

我们提出并分析了一种新的马尔可夫链蒙特卡洛算法，该算法可在全维和非全维多形体上生成均匀样本。该算法称为“矩阵命中并运行”（MHAR），是对命中并运行框架的修改。对于$ n ^ {1+ \ frac {1} {3}} \ ll m $而言，MHAR在软O表示法（$ \ SO $）方面的每个样本的渐近成本比之后的现有采样算法低\ textit {热启动}。MHAR旨在利用需要较少计算和内存资源的矩阵乘法例程。我们的测试表明，此实现比\ textit {hitandrun} R包要快得多，尤其是对于较大的尺寸。最后，我们提供了一个基于Pytorch的python库和一个Colab笔记本，该实现可随时部署在具有GPU或仅CPU的体系结构中。