越简单越好：计算CUR分解的随机算法的比较研究
Simpler is better: A comparative study of randomized algorithms for computing the CUR decomposition
摘要：
CUR分解是一种低秩近似技术，它选择给定矩阵的列和行的小子集作为其列和行空间的基础。由于它比传统的基于规范正交基的低秩分解具有许多优点，近年来引起了人们的广泛兴趣。这些特性包括保留稀疏性或非负性等属性、解释数据的能力以及降低存储需求。找到最小化残差范数的骨架集的问题已知是NP难的，但经典的枢转方案，如列枢转QR工作往往在实践中工作得很好。当与随机化降维技术相结合时，经典的基于旋转的方法变得特别有效，并且已经证明能够非常快速地计算大的、潜在稀疏的矩阵的近似CUR分解。另一类用于计算CUR分解的流行算法是基于使用基于杠杆分数的专用概率分布从完整索引集中随机绘制列和行。这种基于采样的技术对于非常大规模的问题特别有吸引力，并且得到理论性能保证的很好支持。这份手稿提供了一个比较研究的各种随机算法计算CUR分解，最近提出的。此外，它提出了一些修改和简化现有的算法，导致更快的执行时间。
英文摘要：
The CUR decomposition is a technique for low-rank approximation that selects small subsets of the columns and rows of a given matrix to use as bases for its column and rowspaces. It has recently attracted much interest, as it has several advantages over traditional low rank decompositions based on orthonormal bases. These include the preservation of properties such as sparsity or non-negativity, the ability to interpret data, and reduced storage requirements. The problem of finding the skeleton sets that minimize the norm of the residual error is known to be NP-hard, but classical pivoting schemes such as column pivoted QR work tend to work well in practice. When combined with randomized dimension reduction techniques, classical pivoting based methods become particularly effective, and have proven capable of very rapidly computing approximate CUR decompositions of large, potentially sparse, matrices. Another class of popular algorithms for computing CUR de-compositions are based on drawing the columns and rows randomly from the full index sets, using specialized probability distributions based on leverage scores. Such sampling based techniques are particularly appealing for very large scale problems, and are well supported by theoretical performance guarantees. This manuscript provides a comparative study of the various randomized algorithms for computing CUR decompositions that have recently been proposed. Additionally, it proposes some modifications and simplifications to the existing algorithms that leads to faster execution times.
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。
