Researches On The FastICA Algorithm And Its Convergence

Independent Component Analysis(Independent Component Analysis,ICA)is a statistical method for separating independent,non-Gaussian source signals from mixed signals and is widely used.Up to now,a large number of methods on ICA have emerged.And the Fast ICA algorithm is one of the most popular methods.In this paper we mainly study the Fast ICA algorithm and its convergence.The specific work can be summarized as follows:Firstly,an improved Fast ICA algorithm based on Tukey M-estimator,called the T-F algorithm,is proposed.If choosing the Tukey M-estimator with good robust performance that does not involve complex operations such as exponential and logarithm and its Influence Function(Influence Function,IF)is bounded as a nonlinear function(Nonlinear Function,NLF),we can improve the robustness of the Fast ICA algorithm.It is proved that there is always parameter ? of Tukey M-estimator,so that the T-F algorithm satisfies the local stability condition for any non-Gaussian source signals.The computer simulation results show that waveform signals and image signals are successfully separated by using the T-F algorithm with ? =4,and the T-F algorithm is more robust and has higher separation accuracy than the other two algorithms based on M-estimator: H-F and M-F.Secondly,the local convergence and consistency of the Fast ICA algorithm are studied.Breaking through the research difficulty for high-order convergence of the Fast ICA algorithm based on non-kurtosis NLF,we study the convergence order in detail,and give the conditions of convergence of the algorithm in 3rd order and 4th order.Further,the T-F algorithm converges at least in 3rd order,when the source signals obey non-Gaussian distribution with 0 kurtosis,converges at least in 4th order.We use more intuitive method to prove that the column vectors of the mixed matrix are the fixed points of the Fast ICA function and reveal relationship between the fixed point set of the Fast ICA function and the extreme point set of the contrast function.The Dirac function is used to construct the Probability Density Function(PDF)of the observed signal.According to the law of strong numbers,the convergence properties of Fast ICA are extended to ones of the Fast ICA algorithm based on sample.On this basis,estimate from the Fast ICA algorithm is proved to be consistent according to the Z-estimator consistency theorem.Computer simulations verify the consistency of estimate from the Fast ICA algorithm.Finally,the complex-valued ICA is studied.We re-derive the nc-Fast ICA algorithm using generalized linear(or linear-conjugate-linear)transform to make its derivation more theoretical,and degenerate to obtain the c-Fast ICA algorithm.The necessary and sufficient condition to be satisfied for the the fixed point(pseudo fixed point)of the c-Fast ICA algorithm is given.Column vectors of the mixed matrix are proved to be fixed points of the c-Fast ICA function.And the relationship between fixed points of the c-Fast ICA function and local minimum values of contrast function is proved by using the orthogonal projection method.Computer simulations verify three attributes of c-Fast ICA and nc-Fast ICA: both algorithms are convergent;the more the number of samples,the better the separation effect;the two algorithms show poor separation effects on Gaussian source signals.