Randomly pivoted Cholesky: Practical approximation of a kernel matrix with few entry evaluations

作者：Yifan Chen, Ethan N. Epperly, Joel A. Tropp, Robert J. Webber · 发表于：Communications on Pure and Applied Mathematics · 年份：2024 · DOI：10.1002/cpa.22234 · 被引用次数：12 · 研究领域：Stochastic Gradient Optimization Techniques、Sparse and Compressive Sensing Techniques、Machine Learning and Algorithms

Abstract The randomly pivoted Cholesky algorithm ( RPCholesky ) computes a factorized rank‐ approximation of an positive‐semidefinite (psd) matrix. RPCholesky requires only entry evaluations and 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.