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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Veröffentlicht in:	IEEE transaction on neural networks and learning systems Jg. 23; H. 3; S. 425 - 438
Hauptverfasser:	Bouboulis, P., Slavakis, K., Theodoridis, S.
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York, NY IEEE 01.03.2012 Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:	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
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Abstract	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.
AbstractList	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. 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.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.
Author	Slavakis, K. Bouboulis, P. Theodoridis, S.
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Keywords	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
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SubjectTerms	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
Title	Adaptive Learning in Complex Reproducing Kernel Hilbert Spaces Employing Wirtinger's Subgradients
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