The spectrum of a kernel matrix significantly depends on the parameter values of the kernel function used to define the kernel matrix. This makes it challenging to design a preconditioner for a regularized kernel matrix that is robust across different parameter values. This paper proposes the Adaptive Factorized Nystr\"om (AFN) preconditioner. The preconditioner is designed for the case where the rank k of the Nystr\"om approximation is large, i.e., for kernel function parameters that lead to kernel matrices with eigenvalues that decay slowly. AFN deliberately chooses a well-conditioned submatrix to solve with and corrects a Nystr\"om approximation with a factorized sparse approximate matrix inverse. This makes AFN efficient for kernel matrices with large numerical ranks. AFN also adaptively chooses the size of this submatrix to balance accuracy and cost.

翻译：核矩阵的谱特性高度依赖于用于定义核矩阵的核函数参数值。这使得设计一种能够针对不同参数值均具有鲁棒性的正则化核矩阵预处理子充满挑战。本文提出自适应因子化Nyström（AFN）预处理子。该预处理子针对Nyström近似秩k较大的情形设计，即适用于核函数参数导致核矩阵特征值衰减缓慢的场景。AFN通过精心选择条件良好的子矩阵进行求解，并利用因子化稀疏近似逆矩阵对Nyström近似进行校正，从而在核矩阵数值秩较大时仍保持高效。此外，AFN能够自适应地调整该子矩阵的规模，以平衡计算精度与成本。