Robust, randomized preconditioning for kernel ridge regression

Abstract: This paper investigates two randomized preconditioning techniques for solving kernel ridge regression (KRR) problems with a medium to large number of data points ($10⁴ \leq N \leq 10<sup>7$), and it introduces two new methods with state-of-the-art performance. The first method, RPCholesky preconditioning, accurately solves the full-data KRR problem in $O(N<sup>2)$ arithmetic operations, assuming sufficiently rapid polynomial decay of the kernel matrix eigenvalues. The second method, KRILL preconditioning, offers an accurate solution to a restricted version of the KRR problem involving $k \ll N$ selected data centers at a cost of $O((N + k²⁾ k \log k)$ operations. The proposed methods solve a broad range of KRR problems, making them ideal for practical applications.