The randomly pivoted partial Cholesky algorithm (RPCholesky) computes a factorized rank-k approximation of an N x N positive-semidefinite (psd) matrix. RPCholesky requires only (k + 1) N entry evaluations and O(k^2 N) additional arithmetic operations, and it can be implemented with just a few lines of code. The method is particularly useful for approximating a kernel matrix. This paper offers a thorough new investigation of the empirical and theoretical behavior of this fundamental algorithm. For matrix approximation problems that arise in scientific machine learning, experiments show that RPCholesky matches or beats the performance of alternative algorithms. Moreover, RPCholesky provably returns low-rank approximations that are nearly optimal. The simplicity, effectiveness, and robustness of RPCholesky strongly support its use in scientific computing and machine learning applications.

翻译：随机枢轴部分Cholesky算法（RPCholesky）可计算N×N半正定（psd）矩阵的因子化秩k近似。该算法仅需（k+1）N次条目评估及O(k²N)次额外算术运算，且仅需数行代码即可实现。该方法在核矩阵近似问题中尤为实用。本文对该基础算法的经验行为与理论特性进行了全新且全面的研究。针对科学机器学习中的矩阵近似问题，实验表明RPCholesky的性能可与替代算法持平甚至更优。此外，RPCholesky被证明可返回近乎最优的低秩近似。其简洁性、高效性及鲁棒性充分支持其在科学计算与机器学习应用中的使用。