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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.22048 |
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| _version_ | 1866915361460322304 |
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| author | Benning, Felix Schölpple, Max David |
| author_facet | Benning, Felix Schölpple, Max David |
| contents | We provide a characterization for the continuous positive definite kernels on $\mathbb R^d$ that are invariant to linear isometries, i.e. invariant under the orthogonal group $O(d)$. Furthermore, we provide necessary and sufficient conditions for these kernels to be strictly positive definite. This class of isotropic kernels is fairly general: First, it unifies stationary isotropic and dot product kernels, and second, it includes neural network kernels that arise from infinite-width limits of neural networks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_22048 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Schoenberg characterization of continuous non-stationary isotropic positive definite kernels Benning, Felix Schölpple, Max David Statistics Theory 33C50, 33C55, 42A82, 42C10, 43A35, 60G15, 68T07 We provide a characterization for the continuous positive definite kernels on $\mathbb R^d$ that are invariant to linear isometries, i.e. invariant under the orthogonal group $O(d)$. Furthermore, we provide necessary and sufficient conditions for these kernels to be strictly positive definite. This class of isotropic kernels is fairly general: First, it unifies stationary isotropic and dot product kernels, and second, it includes neural network kernels that arise from infinite-width limits of neural networks. |
| title | Schoenberg characterization of continuous non-stationary isotropic positive definite kernels |
| topic | Statistics Theory 33C50, 33C55, 42A82, 42C10, 43A35, 60G15, 68T07 |
| url | https://arxiv.org/abs/2506.22048 |