This survey explores modern approaches for computing low-rank approximations of high-dimensional matrices by means of the randomized SVD, randomized subspace iteration, and randomized block Krylov iteration. The paper compares the procedures via theoretical analyses and numerical studies to highlight how the best choice of algorithm depends on spectral properties of the matrix and the computational resources available. Despite superior performance for many problems, randomized block Krylov iteration has not been widely adopted in computational science. This paper strengthens the case for this method in three ways. First, it presents new pseudocode that can significantly reduce computational costs. Second, it provides a new analysis that yields simple, precise, and informative error bounds. Last, it showcases applications to challenging scientific problems, including principal component analysis for genetic data and spectral clustering for molecular dynamics data.

翻译：本文综述了通过随机奇异值分解、随机子空间迭代及随机块Krylov迭代计算高维矩阵低秩逼近的现代方法。文章通过理论分析与数值实验对比各类算法，重点阐明算法的最优选择如何取决于矩阵谱性质及可用计算资源。尽管随机块Krylov迭代在众多问题中表现优越，但其在计算科学领域尚未得到广泛采用。本文从三方面强化该方法的适用性：首先，提出能显著降低计算开销的新型伪代码；其次，给出简洁精确且具信息量的全新误差界分析；最后，展示其在遗传数据主成分分析与分子动力学数据谱聚类等挑战性科学问题中的应用。