Geometric Learning with Positively Decomposable Kernels

Nathaël da Costa (1) , Cyrus Mostajeran (1) , Juan-Pablo Ortega (1) , Salem Said (2)

1 Nanyang Technological University [Singapour]
2 DAO - Données, Apprentissage et Optimisation

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.

Mots clés: kernel methods; geometric learning; Krein spaces; non-positive kernels; invariant kernels; Gaussian kernel; homogeneous spaces; symmetric spaces

Domaines: Théorie [stat.TH]

Nathaël da Costa, Cyrus Mostajeran, Juan-Pablo Ortega, Salem Said. Geometric Learning with Positively Decomposable Kernels. 2023. ⟨hal-04270518⟩