Invariant kernels on Riemannian symmetric spaces: a harmonic-analytic approach
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866913492493139968 |
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| author | Da Costa, Nathael Mostajeran, Cyrus Ortega, Juan-Pablo Said, Salem |
| author_facet | Da Costa, Nathael Mostajeran, Cyrus Ortega, Juan-Pablo Said, Salem |
| contents | This work aims to prove that the classical Gaussian kernel, when defined on a non-Euclidean symmetric space, is never positive-definite for any choice of parameter. To achieve this goal, the paper develops new geometric and analytical arguments. These provide a rigorous characterization of the positive-definiteness of the Gaussian kernel, which is complete but for a limited number of scenarios in low dimensions that are treated by numerical computations. Chief among these results are the L$^{\!\scriptscriptstyle p}$-$\hspace{0.02cm}$Godement theorems (where $p = 1,2$), which provide verifiable necessary and sufficient conditions for a kernel defined on a symmetric space of non-compact type to be positive-definite. A celebrated theorem, sometimes called the Bochner-Godement theorem, already gives such conditions and is far more general in its scope, but is especially hard to apply. Beyond the connection with the Gaussian kernel, the new results in this work lay out a blueprint for the study of invariant kernels on symmetric spaces, bringing forth specific harmonic analysis tools that suggest many future applications. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_19270 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Invariant kernels on Riemannian symmetric spaces: a harmonic-analytic approach Da Costa, Nathael Mostajeran, Cyrus Ortega, Juan-Pablo Said, Salem Machine Learning Differential Geometry 43A35, 43A85, 43A90, 46E22, 53C35, 53Z50 This work aims to prove that the classical Gaussian kernel, when defined on a non-Euclidean symmetric space, is never positive-definite for any choice of parameter. To achieve this goal, the paper develops new geometric and analytical arguments. These provide a rigorous characterization of the positive-definiteness of the Gaussian kernel, which is complete but for a limited number of scenarios in low dimensions that are treated by numerical computations. Chief among these results are the L$^{\!\scriptscriptstyle p}$-$\hspace{0.02cm}$Godement theorems (where $p = 1,2$), which provide verifiable necessary and sufficient conditions for a kernel defined on a symmetric space of non-compact type to be positive-definite. A celebrated theorem, sometimes called the Bochner-Godement theorem, already gives such conditions and is far more general in its scope, but is especially hard to apply. Beyond the connection with the Gaussian kernel, the new results in this work lay out a blueprint for the study of invariant kernels on symmetric spaces, bringing forth specific harmonic analysis tools that suggest many future applications. |
| title | Invariant kernels on Riemannian symmetric spaces: a harmonic-analytic approach |
| topic | Machine Learning Differential Geometry 43A35, 43A85, 43A90, 46E22, 53C35, 53Z50 |
| url | https://arxiv.org/abs/2310.19270 |