Fast Kernel Summation in High Dimensions via Slicing and Fourier Transforms
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909394628771840 |
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| author | Hertrich, Johannes |
| author_facet | Hertrich, Johannes |
| contents | Kernel-based methods are heavily used in machine learning. However, they suffer from $O(N^2)$ complexity in the number $N$ of considered data points. In this paper, we propose an approximation procedure, which reduces this complexity to $O(N)$. Our approach is based on two ideas. First, we prove that any radial kernel with analytic basis function can be represented as sliced version of some one-dimensional kernel and derive an analytic formula for the one-dimensional counterpart. It turns out that the relation between one- and $d$-dimensional kernels is given by a generalized Riemann-Liouville fractional integral. Hence, we can reduce the $d$-dimensional kernel summation to a one-dimensional setting. Second, for solving these one-dimensional problems efficiently, we apply fast Fourier summations on non-equispaced data, a sorting algorithm or a combination of both. Due to its practical importance we pay special attention to the Gaussian kernel, where we show a dimension-independent error bound and represent its one-dimensional counterpart via a closed-form Fourier transform. We provide a run time comparison and error estimate of our fast kernel summations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_08260 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fast Kernel Summation in High Dimensions via Slicing and Fourier Transforms Hertrich, Johannes Numerical Analysis Machine Learning Kernel-based methods are heavily used in machine learning. However, they suffer from $O(N^2)$ complexity in the number $N$ of considered data points. In this paper, we propose an approximation procedure, which reduces this complexity to $O(N)$. Our approach is based on two ideas. First, we prove that any radial kernel with analytic basis function can be represented as sliced version of some one-dimensional kernel and derive an analytic formula for the one-dimensional counterpart. It turns out that the relation between one- and $d$-dimensional kernels is given by a generalized Riemann-Liouville fractional integral. Hence, we can reduce the $d$-dimensional kernel summation to a one-dimensional setting. Second, for solving these one-dimensional problems efficiently, we apply fast Fourier summations on non-equispaced data, a sorting algorithm or a combination of both. Due to its practical importance we pay special attention to the Gaussian kernel, where we show a dimension-independent error bound and represent its one-dimensional counterpart via a closed-form Fourier transform. We provide a run time comparison and error estimate of our fast kernel summations. |
| title | Fast Kernel Summation in High Dimensions via Slicing and Fourier Transforms |
| topic | Numerical Analysis Machine Learning |
| url | https://arxiv.org/abs/2401.08260 |