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  • Research Article
  • Open Access

Sliding Window Generalized Kernel Affine Projection Algorithm Using Projection Mappings

EURASIP Journal on Advances in Signal Processing20082008:735351

Received: 8 October 2007

Accepted: 17 March 2008

Published: 3 April 2008


Very recently, a solution to the kernel-based online classification problem has been given by the adaptive projected subgradient method (APSM). The developed algorithm can be considered as a generalization of a kernel affine projection algorithm (APA) and the kernel normalized least mean squares (NLMS). Furthermore, sparsification of the resulting kernel series expansion was achieved by imposing a closed ball (convex set) constraint on the norm of the classifiers. This paper presents another sparsification method for the APSM approach to the online classification task by generating a sequence of linear subspaces in a reproducing kernel Hilbert space (RKHS). To cope with the inherent memory limitations of online systems and to embed tracking capabilities to the design, an upper bound on the dimension of the linear subspaces is imposed. The underlying principle of the design is the notion of projection mappings. Classification is performed by metric projection mappings, sparsification is achieved by orthogonal projections, while the online system's memory requirements and tracking are attained by oblique projections. The resulting sparsification scheme shows strong similarities with the classical sliding window adaptive schemes. The proposed design is validated by the adaptive equalization problem of a nonlinear communication channel, and is compared with classical and recent stochastic gradient descent techniques, as well as with the APSM's solution where sparsification is performed by a closed ball constraint on the norm of the classifiers.


  • Linear Subspace
  • Projection Mapping
  • Closed Ball
  • Reproduce Kernel Hilbert Space
  • Equalization Problem

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Authors’ Affiliations

Department of Telecommunications Science and Technology, University of Peloponnese, Tripoli, Greece
Department of Informatics and Telecommunications, University of Athens, Ilissia, Greece


© K. Slavakis and S. Theodoridis 2008

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.