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Unformatted text preview: IEEE SIGNAL PROCESSING LETTERS, VOL. 17, NO. 7, JULY 2010 655 Spurious Solution of the Maximum Likelihood Approach to ICA Fei Ge and Jinwen Ma Abstract— For the separation of linear instantaneous mixtures of independent sources, many Independent Component Analysis (ICA) algorithms can learn the separating matrix by optimizing some objective functions derived from various criteria. The Max- imum Likelihood (ML) principle, with hypothesized model pdf’s, provides an objective function which is commonly used. It is gen- erally considered that the ML approach leads to a separating solu- tion as long as the kurtosis signs of the model pdf’s can correspond and equal to those of the sources, respectively, in some order, which is referred to as the one-bit-matching condition. In this letter, we present an experimental analysis on spurious solution of the ML approach and show that spurious maximum of the objective func- tion really exists in certain cases even if the one-bit-matching con- dition is satisfied. Index Terms— Blind source separation, independent component analysis (ICA), maximum likelihood (ML), one-bit-matching con- dition, spurious solution. I. INTRODUCTION F OR the blind separation of instantaneously mixed inde- pendent non-Gaussian signals, the Independent Compo- nent Analysis (ICA)  is a commonly utilized statistical tech- nique, which exploits only the amplitude statistics of signals . Under this model, the observed signals can be represented by an-dimensional random vector , which is simply a linear transformation of vector of the latent source signals that are mutually independent. For simplicity we assume that is square and invertible, then the sources can be reconstructed as , if is a separating matrix, i.e., has only one nonzero entry in each row and in each column. Various approaches can lead to blind source separation, for example, minimizing mutual information (MMI) , , in- formation maximization (Infomax) . If the joint pdf of the sources is known as , the Maximum Like- lihood (ML) approach provides a consistent estimator of , by maximizing the normalized log-likelihood function : (1) under a given set of i.i.d. samples . However, in the case of ICA, in (1) is not known in practical situations. Manuscript received January 09, 2010; revised March 10, 2010; accepted March 13, 2010. Date of publication May 03, 2010; date of current version May 26, 2010. This work was supported by the Ph.D. Programs Foundation of Ministry of Education of China for Grant 20070001042. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Alfred Mertins. The authors are with the Department of Information Science, School of Mathematical Sciences and LMAM, Peking University, Beijing 100871, China (e-mail: [email protected]; [email protected])....
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This note was uploaded on 12/29/2011 for the course ECE 594C taught by Professor Madhow during the Fall '10 term at UCSB.
- Fall '10
- Signal Processing