An Adaptive Factorized Nyström Preconditioner for Regularized Kernel Matrices
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866929306787119104 |
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| author | Zhao, Shifan Xu, Tianshi Huang, Hua Chow, Edmond Xi, Yuanzhe |
| author_facet | Zhao, Shifan Xu, Tianshi Huang, Hua Chow, Edmond Xi, Yuanzhe |
| contents | 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 Nyström (AFN) preconditioner. The preconditioner is designed for the case where the rank k of the Nyström 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 Nyström 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_05460 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An Adaptive Factorized Nyström Preconditioner for Regularized Kernel Matrices Zhao, Shifan Xu, Tianshi Huang, Hua Chow, Edmond Xi, Yuanzhe Numerical Analysis 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 Nyström (AFN) preconditioner. The preconditioner is designed for the case where the rank k of the Nyström 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 Nyström 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. |
| title | An Adaptive Factorized Nyström Preconditioner for Regularized Kernel Matrices |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2304.05460 |