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**Extra resources for Calculus - One Variable**

**Sample text**

And tensors are written as calligraphic letters (A, B, . . ). This notation is consistently used for lower-order parts of a given structure. e. iN ). e. A = [A1 A2 . ]. g. i = 1, 2, . . , I). is the Khatri-Rao or column-wise Kronecker product. 2 Independent Component Analysis Assume the basic linear statistical model Y =M·X +N (2) where Y ∈ R is called the observation vector, X ∈ R the source vector and N ∈ RI additive noise. M ∈ RI×J is the mixing matrix. The goal of Independent Component Analysis is to estimate the mixing matrix M, and/or the source vector X, given only realizations of Y .

G. [1] [2] [3] and [4], statistics based matrices or tensors often have an identical decomposition. This known decomposition is then used through a Jacobi-like algorithm to estimate the socalled mixing matrix. Perhaps one of the most popular algorithms of that kind is given in [1]. It is called JADE and its goal is to joint-diagonalize a set of hermitian matrices. The algorithm in [5] is intended to “joint-diagonalize” a set of complex symmetric matrices. The ICA algorithm in [2] is intended to diagonalize a ﬁxed order (cumulant) tensor.

We have taken the mixing matrix unitary to avoid the whitening step which may degrade the performances. The objective is to emphasize the existence of an optimal parameter λ which allows an optimal performance of the general algorithm. We get into two situations: one with the 5 states source distribution S1 deﬁned as: {−1; −j; 0; β; jβ} with the probabilities β−1 1 1 1 1 2(1+β) ; 2(1+β) ; β ; 2β(1+β) ; 2β(1+β) , and the other one with the 4 states source distribution S2 deﬁned as: {−1; −j; 0; β; j ∗ β} with uniform probabilities.