Embrace rejection: Kernel matrix approximation by accelerated randomly pivoted Cholesky

TL;DR

This paper introduces an accelerated randomly pivoted Cholesky algorithm that significantly speeds up kernel matrix approximation, combining block computations and rejection sampling, with theoretical guarantees and practical applications.

Contribution

It develops a faster version of RPCholesky using block matrix techniques and rejection sampling, improving efficiency for kernel matrix approximation.

Findings

Over 40x faster kernel matrix approximation

Theoretical guarantees for the accelerated method

Successful application to computational chemistry datasets

Abstract

Randomly pivoted Cholesky (RPCholesky) is an algorithm for constructing a low-rank approximation of a positive-semidefinite matrix using a small number of columns. This paper develops an accelerated version of RPCholesky that employs block matrix computations and rejection sampling to efficiently simulate the execution of the original algorithm. For the task of approximating a kernel matrix, the accelerated algorithm can run over $40\times$ faster. The paper contains implementation details, theoretical guarantees, experiments on benchmark data sets, and an application to computational chemistry.