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A Note on Lewicki-Sejnowski Gradient for Learning Overcomplete Representations

zhshhe{at}scut.edu.cn School of Electronics and Information Engineering, South China University of Technology, Guangzhou, 510640, China, and Laboratory for Advanced Brain Signal Processing, RIKEN Brain Science Institute, Wako-shi, Saitama 351-0198, Japan

Shengli Xie

adshlxie{at}scut.edu.cn School of Electronics and Information Engineering, South China University of Technology, Guangzhou, 510640, China

Liqing Zhang

zhang-lq{at}cs.sjtu.edu.cn Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China

Andrzej Cichocki

cia{at}brain.riken.jp Laboratory for Advanced Brain Signal Processing, RIKEN Brain Science Institute, Wako-shi, Saitama 351-0198, Japan; System Research Institute, Polish Academy of Sciences (PAN), Warsaw, Poland; and Department of Electrical Engineering, Warsaw University of Technology, Warsaw, Poland

Overcomplete representations have greater robustness in noise environment and also have greater flexibility in matching structure in the data. Lewicki and Sejnowski (2000) proposed an efficient extended natural gradient for learning the overcomplete basis and developed an overcomplete representation approach. However, they derived their gradient by many approximations, and their proof is very complicated. To give a stronger theoretical basis, we provide a brief and more rigorous mathematical proof for this gradient in this note. In addition, we propose a more robust constrained Lewicki-Sejnowski gradient.