In addition to recent developments in computing speed and memory, methodological advances have contributed to significant gains in the performance of stochastic simulation. In this paper, we focus on variance reduction for matrix computations via matrix factorization. We provide insights into existing variance reduction methods for estimating the entries of large matrices. Popular methods do not exploit the reduction in variance that is possible when the matrix is factorized. We show how computing the square root factorization of the matrix can achieve in some important cases arbitrarily better stochastic performance. In addition, we propose a factorized estimator for the trace of a product of matrices and numerically demonstrate that the estimator can be up to 1,000 times more efficient on certain problems of estimating the log-likelihood of a Gaussian process. Additionally, we provide a new estimator of the log-determinant of a positive semi-definite matrix where the log-determinant is treated as a normalizing constant of a probability density.

翻译：除计算速度与内存容量的近期进展外，方法论层面的突破亦显著提升了随机模拟的性能。本文聚焦于通过矩阵分解实现矩阵计算的方差缩减技术。我们深入剖析了现有大规模矩阵元素估计方差缩减方法的本质特性后发现，当矩阵可被分解时，主流方法未能充分挖掘其可实现的方差缩减潜力。通过论证平方根矩阵分解在某些重要场景中能够实现任意程度的随机性能提升，我们进一步提出了一种用于矩阵乘积迹估计的因子化估计器。数值实验表明，该估计器在特定高斯过程对数似然估计问题中的计算效率可提升至传统方法的千倍量级。此外，针对正半定矩阵的对数行列式估计，我们创新性地将其视为概率密度函数的归一化常数，并据此构建了新型估计器。