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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.04780 |
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| _version_ | 1866911194034470912 |
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| author | Wortsman, Arie Loureiro, Bruno |
| author_facet | Wortsman, Arie Loureiro, Bruno |
| contents | In this work, we investigate high-dimensional kernel ridge regression (KRR) on i.i.d. Gaussian data with anisotropic power-law covariance. This setting differs fundamentally from the classical source & capacity conditions for KRR, where power-law assumptions are typically imposed on the kernel eigen-spectrum itself. Our contributions are twofold. First, we derive an explicit characterization of the kernel spectrum for polynomial inner-product kernels, giving a precise description of how the kernel eigen-spectrum inherits the data decay. Second, we provide an asymptotic analysis of the excess risk in the high-dimensional regime for a particular kernel with this spectral behavior, showing that the sample complexity is governed by the effective dimension of the data rather than the ambient dimension. These results establish a fundamental advantage of learning with power-law anisotropic data over isotropic data. To our knowledge, this is the first rigorous treatment of non-linear KRR under power-law data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_04780 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Kernel ridge regression under power-law data: spectrum and generalization Wortsman, Arie Loureiro, Bruno Machine Learning In this work, we investigate high-dimensional kernel ridge regression (KRR) on i.i.d. Gaussian data with anisotropic power-law covariance. This setting differs fundamentally from the classical source & capacity conditions for KRR, where power-law assumptions are typically imposed on the kernel eigen-spectrum itself. Our contributions are twofold. First, we derive an explicit characterization of the kernel spectrum for polynomial inner-product kernels, giving a precise description of how the kernel eigen-spectrum inherits the data decay. Second, we provide an asymptotic analysis of the excess risk in the high-dimensional regime for a particular kernel with this spectral behavior, showing that the sample complexity is governed by the effective dimension of the data rather than the ambient dimension. These results establish a fundamental advantage of learning with power-law anisotropic data over isotropic data. To our knowledge, this is the first rigorous treatment of non-linear KRR under power-law data. |
| title | Kernel ridge regression under power-law data: spectrum and generalization |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2510.04780 |