Sliding Window Generalized Kernel Affine Projection Algorithm Using Projection Mappings
- PDF / 611,202 Bytes
- 16 Pages / 600.05 x 792 pts Page_size
- 25 Downloads / 205 Views
Research Article Sliding Window Generalized Kernel Affine Projection Algorithm Using Projection Mappings Konstantinos Slavakis1 and Sergios Theodoridis2 1 Department 2 Department
of Telecommunications Science and Technology, University of Peloponnese, Karaiskaki St., Tripoli 22100, Greece of Informatics and Telecommunications, University of Athens, Ilissia, Athens 15784, Greece
Correspondence should be addressed to Konstantinos Slavakis, [email protected] Received 8 October 2007; Revised 25 January 2008; Accepted 17 March 2008 Recommended by Theodoros Evgeniou 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. Copyright © 2008 K. Slavakis and S. Theodoridis. 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.
1.
INTRODUCTION
Kernel methods play a central role in modern classification and nonlinear regression tasks and they can be viewed as the nonlinear counterparts of linear supervised and unsupervised learning algorithms [1–3]. They are used in a wide variety of applications from pattern analysis [1–3], equalization or identification in communication systems [4, 5], to time series analysis and probability density estimation [6–8]. A positive-definite kernel function defines a high- or even infinite-dimensional reproducing kernel Hilbert space (RKHS) H , widely called feature space [1–3, 9, 10]. It also gives a way to map data, collected from the Euclidean data space
Data Loading...
