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The randomly pivoted partial Cholesky algorithm (RPCholesky) computes afactorized 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) additionalarithmetic 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 andtheoretical behavior of this fundamental algorithm. For matrix approximationproblems that arise in scientific machine learning, experiments show thatRPCholesky matches or beats the performance of alternative algorithms.Moreover, RPCholesky provably returns low-rank approximations that are nearlyoptimal. The simplicity, effectiveness, and robustness of RPCholesky stronglysupport its use in scientific computing and machine learning applications.

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