用随机性寻找结构：构建近似矩阵分解的概率算法
Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions
AI总结：
本文综述并扩展了随机化技术在低秩矩阵近似中的应用，提出模块化框架，通过随机采样识别子空间并压缩矩阵，实现高效、准确、鲁棒的部分矩阵分解。
摘要：
低秩矩阵近似，如截断奇异值分解和秩揭示QR分解，在数据分析和科学计算中扮演核心角色。本文综述并扩展了近期研究，表明随机化为执行低秩矩阵近似提供了强大工具。这些技术比经典方法更充分地利用现代计算架构，并开辟了处理真正大规模数据集的可能性。本文提出了一个模块化框架，用于构建计算部分矩阵分解的随机算法。这些方法使用随机采样来识别捕获矩阵大部分作用的子空间。然后，输入矩阵被显式或隐式地压缩到该子空间，并对缩减后的矩阵进行确定性操作以获得所需的低秩分解。在许多情况下，这种方法在精度、速度和鲁棒性方面优于其经典竞争对手。这些主张得到了大量数值实验和详细误差分析的支持。
英文摘要：
Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed---either explicitly or implicitly---to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, speed, and robustness. These claims are supported by extensive numerical experiments and a detailed error analysis.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
