Quaternion reproducing kernel hilbert spaces: Existence and uniqueness conditions

Tobar F.A.; Mandic D.P.

Abstract

The existence and uniqueness conditions of quaternion reproducing kernel Hilbert spaces (QRKHS) are established in order to provide a mathematical foundation for the development of quaternion-valued kernel learning algorithms. This is achieved through a rigorous account of left quaternion Hilbert spaces, which makes it possible to generalise standard RKHS to quaternion RKHS. Quaternion versions of the Riesz representation and Moore-Aronszajn theorems are next introduced, thus underpinning kernel estimation algorithms operating on quaternion-valued feature spaces. The difference between the proposed quaternion kernel concept and the existing real and vector approaches is also established in terms of both theoretical advantages and computational complexity. The enhanced estimation ability of the so-introduced quaternion-valued kernels over their real- and vector-valued counterparts is validated through kernel ridge regression applications. Simulations on real world 3D inertial body sensor data and nonlinear channel equalisation using novel quaternion cubic and Gaussian kernels support the approach. © 1963-2012 IEEE.

Más información

Título según SCOPUS: Quaternion reproducing kernel hilbert spaces: Existence and uniqueness conditions
Título de la Revista: IEEE TRANSACTIONS ON INFORMATION THEORY
Volumen: 60
Número: 9
Editorial: IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
Fecha de publicación: 2014
Página de inicio: 5736
Página final: 5749
Idioma: English
DOI:

10.1109/TIT.2014.2333734

Notas: SCOPUS