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arXiv · 2304.05460

An Adaptive Factorized Nystr\"om Preconditioner for Regularized Kernel Matrices

Shifan Zhao·Tianshi Xu·Hua Huang·Edmond Chow·Yuanzhe Xi
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Abstract

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.

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Shifan Zhao, Tianshi Xu, Hua Huang, Edmond Chow, Yuanzhe Xi. 2023-04-11. An Adaptive Factorized Nystr\"om Preconditioner for Regularized Kernel Matrices. https://arxiv.org/abs/2304.05460

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