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Unformatted text preview: IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 4, APRIL 2006 1189 Performance Analysis of the FastICA Algorithm and CramrRao Bounds for Linear Independent Component Analysis Petr Tichavsk , Senior Member, IEEE , Zbynek Koldovsk , Member, IEEE , and Erkki Oja , Fellow, IEEE Abstract The FastICA or fixed-point algorithm is one of the most successful algorithms for linear independent component anal- ysis (ICA) in terms of accuracy and computational complexity. Two versions of the algorithm are available in literature and software: a one-unit (deflation) algorithm and a symmetric algorithm. The main result of this paper are analytic closed-form expressions that characterize the separating ability of both versions of the algorithm in a local sense, assuming a good initialization of the algorithms and long data records. Based on the analysis, it is possible to com- bine the advantages of the symmetric and one-unit version algo- rithms and predict their performance. To validate the analysis, a simple check of saddle points of the cost function is proposed that allows to find a global minimum of the cost function in almost 100% simulation runs. Second, the CramrRao lower bound for linear ICA is derived as an algorithm independent limit of the achievable separation quality. The FastICA algorithm is shown to approach this limit in certain scenarios. Extensive computer simulations sup- porting the theoretical findings are included. Index Terms Blind source separation, independent component analysis (ICA), CramrRao lower bound. I. INTRODUCTION B LIND SOURCE separation (BSS), which consists of recovering original signals from their mixtures when the mixing process is unknown, has been a widely studied problem in signal processing for the last two decades (for a review, see ). Independent component analysis (ICA), a statistical method for signal separation , , is also a well-known issue in the community. Its aim is to transform the mixed random signals into source signals or components that are as mutually Manuscript received November 26, 2004; revised May 19, 2005. The work of P. Tichavsk was supported by Ministry of Education, Youth and Sports of the Czech Republic through the project 1M0572. Part of this paper (on the CRB) was presented at International Conference on Acoustics, Speech and Signal Pro- cessing (ICASSP), Philadelphia, PA, 2005, and another part (analysis of Fas- tICA) was presented at the 13th IEEE/SP Statistical Signal Processing Work- shop, Bordeaux, France, July, 1720, 2005. The associate editor coordinating the review of this paper and approving it for publication was Prof. Jonathon Chambers. P. Tichavsk is with the Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, P 182 08 Prague 8, Czech Republic (e-mail: email@example.com; website: http://si.utia.cas.cz/Tichavsky.html)....
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