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arXiv · 2310.13821

Geometric Learning with Positively Decomposable Kernels

Nathael Da Costa·Cyrus Mostajeran·Juan-Pablo Ortega·Salem Said
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Abstract

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.

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Nathael Da Costa, Cyrus Mostajeran, Juan-Pablo Ortega, Salem Said. 2024-07-30. Geometric Learning with Positively Decomposable Kernels. https://arxiv.org/abs/2310.13821

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