Geometric Learning with Positively Decomposable Kernels
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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_ | 1866909273074696192 |
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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 | Kernel methods are powerful tools in machine learning. Classical kernel methods are based on positive-definite kernels, which map data spaces into reproducing kernel Hilbert spaces (RKHS). For non-Euclidean data spaces, positive-definite kernels are difficult to come by. In this case, we propose the use of reproducing kernel Krein space (RKKS) based methods, which require only kernels that admit a positive decomposition. We show that one does not need to access this decomposition in order to learn in RKKS. We then investigate the conditions under which a kernel is positively decomposable. We show that invariant kernels admit a positive decomposition on homogeneous spaces under tractable regularity assumptions. This makes them much easier to construct than positive-definite kernels, providing a route for learning with kernels for non-Euclidean data. By the same token, this provides theoretical foundations for RKKS-based methods in general. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_13821 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Geometric Learning with Positively Decomposable Kernels Da Costa, Nathael Mostajeran, Cyrus Ortega, Juan-Pablo Said, Salem Machine Learning Differential Geometry Kernel methods are powerful tools in machine learning. Classical kernel methods are based on positive-definite kernels, which map data spaces into reproducing kernel Hilbert spaces (RKHS). For non-Euclidean data spaces, positive-definite kernels are difficult to come by. In this case, we propose the use of reproducing kernel Krein space (RKKS) based methods, which require only kernels that admit a positive decomposition. We show that one does not need to access this decomposition in order to learn in RKKS. We then investigate the conditions under which a kernel is positively decomposable. We show that invariant kernels admit a positive decomposition on homogeneous spaces under tractable regularity assumptions. This makes them much easier to construct than positive-definite kernels, providing a route for learning with kernels for non-Euclidean data. By the same token, this provides theoretical foundations for RKKS-based methods in general. |
| title | Geometric Learning with Positively Decomposable Kernels |
| topic | Machine Learning Differential Geometry |
| url | https://arxiv.org/abs/2310.13821 |