Adaptive Learning in Complex Reproducing Kernel Hilbert Spaces Employing Wirtinger's Subgradients

This paper presents a wide framework for non-linear online supervised learning tasks in the context of complex valued signal processing. The (complex) input data are mapped into a complex reproducing kernel Hilbert space (RKHS), where the learning phase is taking place. Both pure complex kernels and...

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Bibliographic Details
Published in:IEEE transaction on neural networks and learning systems Vol. 23; no. 3; pp. 425 - 438
Main Authors: Bouboulis, P., Slavakis, K., Theodoridis, S.
Format: Journal Article
Language:English
Published: New York, NY IEEE 01.03.2012
Institute of Electrical and Electronics Engineers
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Adaptive kernel learning
Applied sciences
Artificial intelligence
Calculus
Channels
complex kernels
Computer science; control theory; systems
Context
Equalization
Exact sciences and technology
Hilbert space
Kernel
Kernels
Learning
Learning and adaptive systems
Loss measurement
Machine learning
Mathematical analysis
Nonlinearity
On-line systems
projection
Signal processing
Studies
subgradient
Tasks
widely linear estimation
Wirtinger's calculus
complex kernels
subgradient
Non linear channel
Complex signal
Equalization
widely linear estimation
Adaptive method
Quadrature phase shift keying
Kernel method
Complex variable method
Loss function
Supervised learning
Efficiency
Linear filter
Signal processing
Sparse representation
Signal sources
Adaptive kernel learning
Hilbert space
Linear estimation
projection
Wirtinger's calculus
Learning algorithm
ISSN:2162-237X, 2162-2388
Online Access:Get full text
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Summary:This paper presents a wide framework for non-linear online supervised learning tasks in the context of complex valued signal processing. The (complex) input data are mapped into a complex reproducing kernel Hilbert space (RKHS), where the learning phase is taking place. Both pure complex kernels and real kernels (via the complexification trick) can be employed. Moreover, any convex, continuous and not necessarily differentiable function can be used to measure the loss between the output of the specific system and the desired response. The only requirement is the subgradient of the adopted loss function to be available in an analytic form. In order to derive analytically the subgradients, the principles of the (recently developed) Wirtinger's calculus in complex RKHS are exploited. Furthermore, both linear and widely linear (in RKHS) estimation filters are considered. To cope with the problem of increasing memory requirements, which is present in almost all online schemes in RKHS, the sparsification scheme, based on projection onto closed balls, has been adopted. We demonstrate the effectiveness of the proposed framework in a non-linear channel identification task, a non-linear channel equalization problem and a quadrature phase shift keying equalization scheme, using both circular and non circular synthetic signal sources.
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ISSN:2162-237X
2162-2388
DOI:10.1109/TNNLS.2011.2179810