Mathematics in Independent Component Analysis

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86 Chapter 3. Signal Processing 84(5):951-956, 2004 the preceding section—here dependence is allowed between real-valued couples of random variables. Let k; n ∈ N such that k divides n. We call an n-dimensional random vector Y k-independent if the k-dimensional random vectors ⎛ ⎞ ⎛ ⎞ ⎜ ⎝ Y1 . Yk ⎟ ⎠ ;:::; ⎜ ⎝ Yn−k+1 . Yn ⎟ ⎠ are mutually independent. A matrix W ∈ Gl(n; R) is called a k-multidimensional ICA of an n-dimensional random vector X if WX is k-independent. If k =1, this is the same as ordinary ICA. Obvious indeterminacies are, similar to ordinary ICA, invertible transforms in Gl(k; R) in each tuple as well as the fact that the order of the independent k-tuples is not xed. So, de ne for r; s =1;:::;n=k the (r; s) sub-k-matrix of W =(wij) tobethek × k submatrix (wij) i=rk; :::; rk+k−1 j=sk; :::; sk+k−1 that is the k × k submatrix of W starting at position (rk; sk). A matrix L ∈ Gl(n; R) is said to be a k-scaling and permutation matrix if for each r=1;:::;n=kthere exists precisely one s with the (r; s) sub-k-matrix of L to be nonzero, and such that this submatrix is in Gl(k; R), and if for each s=1;:::;n=kthere exists only one r with the (r; s) sub-k-matrix satisfying the same condition. Hence, if Y is k-independent, also LY is k-independent. Two matrices A and B are said to be k-equivalent, A ∼k B, if there exists such a k-scaling and permutation matrix L with A=BL. As stated above, given two matrices W and V with W −1 ∼k V −1 such that one of them is a k-multidimensional ICA of a given random vector, then so is the other. We will show that there are no more indeterminacies of multidimensional ICA. As usual multidimensional ICA can solve the multidimensional BSS problem X = AS; where A ∈ Gl(n; R) and S is a k-independent n-dimensional random vector. Finding the indeterminacies of multidimensional ICA then shows that A can be found except for k-equivalence (separability), because if X = AS and W is a demixing matrix F.J. Theis / Signal Processing 84 (2004) 951 – 956 955 such that WX is k-independent, then WA ∼k I, so W−1 ∼k A as desired. However, for the proof we need one more condition for A: We call A k-admissible if for each r; s = 1;:::;n=kthe (r; s) sub-k-matrix of A is either invertible or zero. Note that this is not a strong restriction— if we randomly choose A with coe cients out of a continuous distribution, then with probability one we get a k-admissible matrix, because the non-k-admissible matrices ⊂ Rn2 lie in a submanifold of dimension smaller than n2 . Theorem 5.1 (Separability of multidimensional BSS): Let A ∈ Gl(n; R) and S a k-independent n-dimensional random vector having no Gaussian k-tuple (Srk;:::;Srk+k−1) T . Assume that A is k-admissible. If AS is again k-independent, then A is k-equivalent to the identity. For the case k = 1 this is linear BSS separability because every matrix is 1-admissible. Proof. Denote X := AS. Assume that A k I. Then there exist indices r1;r2 and s such that the (r1;s) and the (r2;s) sub-k-matrices of A are non-zero (hence in Gl(k; R) by k-admissability). Applying Corollary 3.3 to the two random vectors (Xr1k;:::;Xr1k+k−1) T and (Xr2k;:::;Xr2k+k−1) T then shows that (Ssk;:::;Ssk+k−1) T is Gaussian, which is a contradiction. Note that we could have used whitening to assume that A is orthogonal; however there does not seem to be a direct way to exploit this in order to allow one fully Gaussian k-tuple, contrary to the complex ICA case, see Theorem 4.1. 6. Conclusion Uniqueness and separability results play a central role in solving BSS problems since they allow algorithms to apply ICA in order to uniquely (except for scaling and permutation) nd the original mixing matrices. We have used a multidimensional version of the Skitovitch–Darmois theorem in order to calculate the
Chapter 3. Signal Processing 84(5):951-956, 2004 87 956 F.J. Theis / Signal Processing 84 (2004) 951 – 956 indeterminacies of complex and of multidimensional ICA. In the multidimensional ICA case an additional restriction was needed in the proof, which could be relaxed if Corollary 3.3 can be extended to allow matrices of arbitrary rank. Acknowledgements This research was supported by Grants from the DFG 2 and the BMBF 3 . References [1] A. Back, A. Tsoi, Blind deconvolution of signals using a complex recurrent network, in: Neural Networks for Signal Processing 4, Proceedings of the 1994 IEEE Workshop, 1994, pp. 565–574. [2] E. Bingham, A. Hyvarinen, A fast xed-point algorithm for independent component analysis of complex-valued signals, Internat. J. Neural Systems 10 (1) (2000) 1–8. [3] J. Cardoso, Multidimensional independent component analysis, in: Proceedings of ICASSP ’98, Seattle, WA, May 12–15, 1998. [4] A. Cichocki, S. Amari, Adaptive Blind Signal and Image Processing, Wiley, New York, 2002. [5] P. Comon, Independent component analysis—a new concept? Signal Processing 36 (1994) 287–314. [6] G. Darmois, Analyse generale des liaisons stochastiques, Rev. Inst. Internat. Statist. 21 (1953) 2–8. 2 Graduate college ‘nonlinear dynamics’. 3 Project ‘ModKog’. [7] S. Fiori, Blind separation of circularly distributed sources by neural extended apex algorithm, Neurocomputing 34 (2000) 239–252. [8] S. Ghurye, I. Olkin, A characterization of the multivariate normal distribution, Ann. Math. Statist. 33 (1962) 533–541. [9] A. Hyvarinen, P. Hoyer, Emergence of phase and shift invariant features by decomposition of natural images into independent feature subspaces, Neural Computation 12 (7) (2000) 1705–1720. [10] A. Hyvarinen, P. Hoyer, M. Inki, Topographic independent component analysis, Neural Computation 13 (7) (2001) 1525–1558. [11] A. Hyvarinen, J. Karhunen, E. Oja, Independent Component Analysis, Wiley, New York, 2001. [12] A. Kagan, Y. Linnik, C. Rao, Characterization Problems in Mathematical Statistics, Wiley, New York, 1973. [13] E. Moreau, O. Macchi, Higher order contrasts for self-adaptive source separation, Internat. J. Adaptive Control Signal Process. 10 (1) (1996) 19–46. [14] V. Skitovitch, On a property of the normal distribution, DAN SSSR 89 (1953) 217–219. [15] V. Skitovitch, Linear forms in independent random variables and the normal distribution law, Izvestiia AN SSSR, Ser. Matem. 18 (1954) 185–200. [16] P. Smaragdis, Blind separation of convolved mixtures in the frequency domain, Neurocomputing 22 (1998) 21–34. [17] F. Theis, A new concept for separability problems in blind source separation, 2003, submitted for publication; preprint at http://homepages.uni-regensburg.de/ ∼ thf11669/publications/ preprints/theis03linuniqueness.pdf [18] A. Zinger, Investigations into analytical statistics and their application to limit theorems of probability theory, Ph.D. Thesis, Leningrad University, 1969.
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Mathematics in Independent Component Analysis

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