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Research Article A Unified View of Adaptive Variable-Metric Projection Algorithms | Hindawi Publishing Corporation EURASIP Journal on Advances in Signal Processing Volume 2009 Article ID 589260 13 pages doi 2009 589260 Research Article A Unified View of Adaptive Variable-Metric Projection Algorithms Masahiro Yukawa1 and Isao Yamada2 1 Mathematical Neuroscience Laboratory BSI RIKEN 2-1 Hirosawa Wako Saitama 351-0198 Japan 2 Department of Communications and Integrated Systems Tokyo Institute of Technology Meguro-ku Tokyo 152-8552 Japan Correspondence should be addressed to Masahiro Yukawa myukawa@ Received 24 June 2009 Accepted 29 October 2009 Recommended by Vitor Nascimento We present a unified analytic tool named variable-metric adaptive projected subgradient method V-APSM that encompasses the important family of adaptive variable-metric projection algorithms. The family includes the transform-domain adaptive filter the Newton-method-based adaptive filters such as quasi-Newton the proportionate adaptive filter and the Krylov-proportionate adaptive filter. We provide a rigorous analysis of V-APSM regarding several invaluable properties including monotone approximation which indicates stable tracking capability and convergence to an asymptotically optimal point. Small metric-fluctuations are the key assumption for the analysis. Numerical examples show i the robustness of V-APSM against violation of the assumption and ii the remarkable advantages over its constant-metric counterpart for colored and nonstationary inputs under noisy situations. Copyright 2009 M. Yukawa and I. Yamada. 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 The adaptive projected subgradient method APSM 13 serves as a unified guiding principle of many existing projection algorithms including the normalized least mean square NLMS algorithm 4 5 the affine projection algorithm APA 6 7 the .