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. The 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：本综述探讨了通过随机化奇异值分解、随机化子空间迭代和随机化块Krylov迭代计算高维矩阵低秩逼近的现代方法。论文通过理论分析与数值研究比较了各方法的性能，揭示了最优算法选择取决于矩阵谱特性与可用计算资源。尽管随机化块Krylov迭代在众多问题中表现卓越，但尚未在计算科学领域得到广泛应用。论文从三个方面强化了该方法的适用性论证：首先，提出能显著降低计算成本的新型伪代码；其次，给出简洁、精确且具有指导意义的误差界新分析；最后，展示了其在典型科学问题（包括遗传数据主成分分析与分子动力学数据谱聚类）中的应用。