Conference Paper
Convergence of the symmetrical FastICA algorithm
Neural Networks Res. Centre, Helsinki Univ. of Technol., Espoo, Finland
DOI: 10.1109/ICONIP.2002.1202844 Conference: Neural Information Processing, 2002. ICONIP '02. Proceedings of the 9th International Conference on, Volume: 3 Source: IEEE Xplore
ABSTRACT
The FastICA algorithm is one of the most popular methods to solve problems in independent component analysis (ICA) and blind source separation. It has been shown experimentally that it outperforms most of the commonly used ICA algorithms in convergence speed. A rigorous convergence analysis has been presented only for the socalled oneunit case, in which just one of the rows of the separating matrix is considered. However, in the FastICA algorithm, there is also an explicit normalization step, and it may be questioned whether the extra rotation caused by the normalization will effect the convergence speed. The purpose of this paper is to show that this is not the case and the good convergence properties of the oneunit case are also shared by the full algorithm with symmetrical normalization.

 "Hence, we are in front of two problems: 1) the convergence and the convergence rate of the algorithm, 2) the identification of the columns of the mixing matrix among the set of the fixed points. For the kurtosis, the most widely used contrast function both in FastICA and in many other ICA algorithms [8], [9], the convergence rate is cubic, as shown in [6], [10], [11]. For a general contrast function, the analysis of the convergence rate was done in [3], [7], [12], showing quadratic convergence rate. "
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ABSTRACT: Using an infinite sample, the contrast function and the FastICA algorithm are deterministic. In the practical case, we have only a finite sample. Then the contrast function and the FastICA algorithm become estimators of the deterministic case. This paper provides a unified study of the deflation FastICA algorithm assuming a finite or an infinite sample. We consider four random probability distributions based on the finite sample, and construct four FastICA estimators. We show that under mild conditions, each of these estimators are equal to a local minimizer of the contrast function with respect to the underlying random probability distribution. Making use of the existing results of Mestimators, we give a rigorous analysis of the asymptotic errors of FastICA estimators. We derive five criteria for the optimal choice of the nonlinearity function. 
 "Thus one arrives at a gradient descent method for determining the ICA basis, and since this can be phrased as a fixedpoint problem, mechanisms similar to the Contraction Mapping Theorem can be brought to bear for existence, uniqueness, and convergence rate. The method we use here is the FastICA algorithm [14]. "
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ABSTRACT: We investigate the implications of a unified spatiochromatic basis for image compression and reconstruction. Different adaptive and general methods (principal component analysis, PCA, independent component analysis, ICA, and discrete cosine transform, DCT) are applied to generate bases. While typically such bases with spatial extent are investigated in terms of their correspondence to human visual perception, we are interested in their applicability to multimedia encoding. The performance of the extracted spatiochromatic spatial patch bases is evaluated in terms of quality of reconstruction with respect to their potential for data compression. Since ICA is not as widely used as it should be, compared to the other decorrelation methods applied here in a new domain, we also provide a review of ICA. The results discussed here are intended to provide another path towards perceptually based encoding of visual data. This leads to a deeper understanding of the role played by chromatic features in data reduction. 
 "A socalled symmetric FastICA algorithm, a parallel version of FastICA on the orthogonal group, has been developed to recover all source signals simultaneously [3]. The local convergence properties of the symmetric FastICA were firstly discussed with a special case in [11]. More recently, the problem has been further investigated under a general setting by Oja and Yuan in [12]. "
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ABSTRACT: The FastICA algorithm is one of the most prominent methods to solve the problem of linear independent component analysis (ICA). Although there have been several attempts to prove local convergence properties of FastICA, rigorous analysis is still missing in the community. The major difficulty of analysis is because of the wellknown signflipping phenomenon of FastICA, which causes the discontinuity of the corresponding FastICA map on the unit sphere. In this paper, by using the concept of principal fiber bundles, FastICA is proven to be locally quadratically convergent to a correct separation. Higher order local convergence properties of FastICA are also investigated in the framework of a scalar shift strategy. Moreover, as a parallelized version of FastICA, the socalled QR FastICA algorithm, which employs the QR decomposition (GramSchmidt orthonormalization process) instead of the polar decomposition, is shown to share similar local convergence properties with the original FastICA.
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