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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.03608 |
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| _version_ | 1866910894936555520 |
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| author | Kielstra, P. Michael Lindsey, Michael |
| author_facet | Kielstra, P. Michael Lindsey, Michael |
| contents | We introduce a fast algorithm for Gaussian process regression in low dimensions, applicable to a widely-used family of non-stationary kernels. The non-stationarity of these kernels is induced by arbitrary spatially-varying vertical and horizontal scales. In particular, any stationary kernel can be accommodated as a special case, and we focus especially on the generalization of the standard Matérn kernel. Our subroutine for kernel matrix-vector multiplications scales almost optimally as $O(N\log N)$, where $N$ is the number of regression points. Like the recently developed equispaced Fourier Gaussian process (EFGP) methodology, which is applicable only to stationary kernels, our approach exploits non-uniform fast Fourier transforms (NUFFTs). We offer a complete analysis controlling the approximation error of our method, and we validate the method's practical performance with numerical experiments. In particular we demonstrate improved scalability compared to to state-of-the-art rank-structured approaches in spatial dimension $d>1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_03608 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gaussian process regression with log-linear scaling for common non-stationary kernels Kielstra, P. Michael Lindsey, Michael Numerical Analysis Computation We introduce a fast algorithm for Gaussian process regression in low dimensions, applicable to a widely-used family of non-stationary kernels. The non-stationarity of these kernels is induced by arbitrary spatially-varying vertical and horizontal scales. In particular, any stationary kernel can be accommodated as a special case, and we focus especially on the generalization of the standard Matérn kernel. Our subroutine for kernel matrix-vector multiplications scales almost optimally as $O(N\log N)$, where $N$ is the number of regression points. Like the recently developed equispaced Fourier Gaussian process (EFGP) methodology, which is applicable only to stationary kernels, our approach exploits non-uniform fast Fourier transforms (NUFFTs). We offer a complete analysis controlling the approximation error of our method, and we validate the method's practical performance with numerical experiments. In particular we demonstrate improved scalability compared to to state-of-the-art rank-structured approaches in spatial dimension $d>1$. |
| title | Gaussian process regression with log-linear scaling for common non-stationary kernels |
| topic | Numerical Analysis Computation |
| url | https://arxiv.org/abs/2407.03608 |