Independent Component Analysis (ICA) - one of the basic tools in data analysis - aims to find a coordinate system in which the components of the data are independent. Most popular ICA methods use kurtosis as a metric of non-Gaussianity to maximize, such as FastICA and JADE. However, their assumption of fourth-order moment (kurtosis) may not always be satisfied in practice. One of the possible solution is to use third-order moment (skewness) instead of kurtosis, which was applied in @math and EcoICA. In this paper we present a competitive approach to ICA based on the Split Generalized Gaussian distribution (SGGD), which is well adapted to heavy-tailed as well as asymmetric data. Consequently, we obtain a method which works better than the classical approaches, in both cases: heavy tails and non-symmetric data.
Lacoume and Ruiz @cite_55 sketched a mathematical approach to the problem using higher-order statistics, which can be interpreted as a measure of fitting independent components. Cardoso @cite_62 @cite_48 focused on the algebraic properties of the fourth-order cumulant (kurtosis), which is still a popular approach @cite_24 .
@cite_24 @cite_48 @cite_55 @cite_62
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Comon's (1994) well-known scheme for independent component analysis (ICA) is based on the maximal diagonalization, in a least-squares sense, of a higher-order cumulant tensor. In a previous paper, we proved that for fourth-order cumulants, the computation of an elementary Jacobi rotation is equivalent to the computation of the best rank-1 approximation of a fourth-order tensor. In this paper, we show that for third-order tensors, the computation of an elementary Jacobi rotation is again equivalent to a best rank-1 approximation; however, here, it is a matrix that has to be approximated. This favorable computational load makes it attractive to do "something third-order-like" for fourth-order cumulant tensors as well. We show that simultaneous optimal diagonalization of "third-order tensor slices" of the fourth-order cumulant is a suitable strategy. This "simultaneous third-order tensor diagonalization" approach (STOTD) is similar in spirit to the efficient JADE-algorithm. Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253 Ideas for higher-order array processing are introduced, focusing on fourth-order cumulant statistics. They are expressed in an index-free formalism allowing the exploitation of all their symmetry properties. It is shown that, when dealing with 4-index quantities, symmetries are related to rank properties. The rich symmetry structure yields a class of identification algorithms. An algebraic technique for blind identification is included, and a few others are briefly indicated. > This discussion presents a new perspective of subspace independent component analysis (ICA). The notion of a function of cumulants (kurtosis) is generalized to vector kurtosis. This vector kurtosis is utilized in the subspace ICA algorithm to estimate subspace independent components. One of the main advantages of the presented approach is its computational simplicity. The experiments have shown promising results in estimating subspace independent components.