SIAM Review, Volume 64, Issue 1, Page 3-56, February 2022.
Efficient sampling from a high-dimensional Gaussian distribution is an old but high-stakes issue. Vanilla Cholesky samplers imply a computational cost and memory requirements that can rapidly become prohibitive in high dimensions. To tackle these issues, multiple methods have been proposed from different communities ranging from iterative numerical linear algebra to Markov chain Monte Carlo (MCMC) approaches. Surprisingly, no complete review and comparison of these methods has been conducted. This paper aims to review all these approaches by pointing out their differences, close relations, benefits, and limitations. In addition to reviewing the state of the art, this paper proposes a unifying Gaussian simulation framework by deriving a stochastic counterpart of the celebrated proximal point algorithm in optimization. This framework offers a novel and unifying revisiting of most of the existing MCMC approaches while also extending them. Guidelines to choosing the appropriate Gaussian simulation method for a given sampling problem in high dimensions are proposed and illustrated with numerical examples.

中文翻译：

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高维高斯采样：基于随机近点算法的回顾和统一方法

SIAM 评论，第 64 卷，第 1 期，第 3-56 页，2022 年 2 月。
从高维高斯分布中进行有效采样是一个古老但风险很高的问题。Vanilla Cholesky 采样器意味着计算成本和内存要求在高维时会迅速变得令人望而却步。为了解决这些问题，从迭代数值线性代数到马尔可夫链蒙特卡罗 (MCMC) 方法等不同社区提出了多种方法。令人惊讶的是，尚未对这些方法进行完整的审查和比较。本文旨在通过指出它们的差异、密切关系、好处和局限性来回顾所有这些方法。除了回顾现有技术之外，本文还通过推导优化中著名的近点算法的随机对应物，提出了一个统一的高斯模拟框架。该框架提供了对大多数现有 MCMC 方法的新颖和统一的重新审视，同时还扩展了它们。提出了为给定的高维采样问题选择适当高斯模拟方法的指南，并通过数值示例进行了说明。