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SVD、随机化子空间迭代和随机化块Krylov迭代计算高维矩阵低秩近似的现代方法。论文通过理论分析和数值研究对这些方法进行了比较，以突出最佳算法选择如何取决于矩阵的谱性质和可用的计算资源。尽管对于许多问题表现出优越性能，但随机化块Krylov迭代在计算科学中尚未得到广泛应用。本文从三个方面加强了对该方法的论证：首先，提出了能够显著降低计算成本的新伪代码；其次，提供了新的分析，得出了简单、精确且具信息性的误差界；最后，展示了该方法在具有挑战性的科学问题中的应用，包括遗传数据的主成分分析和分子动力学数据的谱聚类。