Funktionentheorie by Hurwitz A , Courant R

By Hurwitz A , Courant R

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81) indicates that an additive constant to a random variable influences only the phase behaviour of the function and the magnitude is not affected. 82) shows that a scale change in the random variable is mirrored by a scale change in the w variable. Finally the sum of two random independent variables has a moment generating function that is the product of the moment generating functions of the individual random variables being summed. Similar properties hold for the cumulant generating function.

By fixing the output to have an identity covariance matrix, Comon reduces the scaling indeterminacy to sign indeterminacy so that if' =LlPF where Ll is a diagonal matrix with entries ±l. The matrix transform F can be defined as F = QLA where LA is a lower triangular matrix and Q is a rotation. The rotation is required to maintain the identity covariance matrix of the final output. So the first part of the algorithm computes the lower triangular matrix LA such that the transformed data z = LAx has an identity covariance C zz = E{ ZZT} =I.

92) i l ,12, .. 41). 92) sits well from an information theoretic perspective. 2 is that presented in (Comon, 1994) reproduced here for completeness of presentation. Proof: The input variable x is transformed by the linear rotation W such that W WT=I, to y, and y=Wx. ,j, and so the following holds due to the multilinearity of cumulants (Brilinger, 1981) /(ijk = L W ip Wjq Wkrlpqr pqr Sr = ~ ~ ~ ~ ~WiP ... J I I Wiq ... 94) As W is orthogonal then LWipWiq =8 pq = i I: p = q { . 96) This theorem has been the basis of many of the batch-based algorithms and techniques which will be considered in Chapter 3.

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