Geometric Learning with Positively Decomposable Kernels - Optimization and learning for Data Science
Pré-Publication, Document De Travail Année : 2024

Geometric Learning with Positively Decomposable Kernels

Résumé

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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Dates et versions

hal-04270518 , version 1 (04-11-2023)
hal-04270518 , version 2 (09-09-2024)

Identifiants

  • HAL Id : hal-04270518 , version 2

Citer

Nathaël da Costa, Cyrus Mostajeran, Juan-Pablo Ortega, Salem Said. Geometric Learning with Positively Decomposable Kernels. 2024. ⟨hal-04270518v2⟩
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