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：核矩阵的谱分布显著依赖于定义该矩阵的核函数参数取值。这给设计对多参数取值具有鲁棒性的正则化核矩阵预处理方法带来挑战。本文提出自适应分解Nyström（AFN）预处理方法。该预处理方法针对Nyström近似秩k较大的情形设计，即针对那些核函数参数导致核矩阵特征值衰减缓慢的案例。AFN通过精心选取良态子矩阵进行求解，并利用因子化稀疏近似矩阵逆修正Nyström近似。这使得AFN对数值秩较大的核矩阵具有高效性。同时，AFN能够自适应调整该子矩阵的规模以平衡精度与计算开销。