Efficient Fourier representations of families of Gaussian processes
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866913375416483840 |
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| author | Greengard, Philip |
| author_facet | Greengard, Philip |
| contents | We introduce a class of algorithms for constructing Fourier representations of Gaussian processes in $1$ dimension that are valid over ranges of hyperparameter values. The scaling and frequencies of the Fourier basis functions are evaluated numerically via generalized quadratures. The representations introduced allow for $O(m^3)$ inference, independent of $N$, for all hyperparameters in the user-specified range after $O(N + m^2\log{m})$ precomputation where $N$, the number of data points, is usually significantly larger than $m$, the number of basis functions. Inference independent of $N$ for various hyperparameters is facilitated by generalized quadratures, and the $O(N + m^2\log{m})$ precomputation is achieved with the non-uniform FFT. Numerical results are provided for Matérn kernels with $ν\in [3/2, 7/2]$ and lengthscale $ρ\in [0.1, 0.5]$ and squared-exponential kernels with lengthscale $ρ\in [0.1, 0.5]$. The algorithms of this paper generalize mathematically to higher dimensions, though they suffer from the standard curse of dimensionality. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2109_14081 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Efficient Fourier representations of families of Gaussian processes Greengard, Philip Computation Numerical Analysis We introduce a class of algorithms for constructing Fourier representations of Gaussian processes in $1$ dimension that are valid over ranges of hyperparameter values. The scaling and frequencies of the Fourier basis functions are evaluated numerically via generalized quadratures. The representations introduced allow for $O(m^3)$ inference, independent of $N$, for all hyperparameters in the user-specified range after $O(N + m^2\log{m})$ precomputation where $N$, the number of data points, is usually significantly larger than $m$, the number of basis functions. Inference independent of $N$ for various hyperparameters is facilitated by generalized quadratures, and the $O(N + m^2\log{m})$ precomputation is achieved with the non-uniform FFT. Numerical results are provided for Matérn kernels with $ν\in [3/2, 7/2]$ and lengthscale $ρ\in [0.1, 0.5]$ and squared-exponential kernels with lengthscale $ρ\in [0.1, 0.5]$. The algorithms of this paper generalize mathematically to higher dimensions, though they suffer from the standard curse of dimensionality. |
| title | Efficient Fourier representations of families of Gaussian processes |
| topic | Computation Numerical Analysis |
| url | https://arxiv.org/abs/2109.14081 |