This paper introduces two randomized preconditioning techniques for robustly solving kernel ridge regression (KRR) problems with a medium to large number of data points ($10^4 \leq N \leq 10^7$). The first method, RPCholesky preconditioning, is capable of accurately solving the full-data KRR problem in $O(N^2)$ 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 $O((N + k^2) k \log k)$ operations. The proposed methods solve a broad range of KRR problems and overcome the failure modes of previous KRR preconditioners, making them ideal for practical applications.

翻译：本文提出了两种随机预条件技术，用于稳健求解中等至大规模数据点（$10^4 \leq N \leq 10^7$）的核岭回归（KRR）问题。第一种方法为RPCholesky预条件，在核矩阵特征值足够快多项式衰减的假设下，能以$O(N^2)$次算术运算精确求解完整数据KRR问题。第二种方法为KRILL预条件，能以$O((N + k^2) k \log k)$次运算精确求解涉及$k \ll N$个选定数据中心点的受限KRR问题。所提方法能求解广泛的KRR问题，并克服了先前KRR预条件器的失效模式，使其成为实际应用的理想选择。