Complex random vectors and ICA models: Identifiability, uniqueness, and separability

IEEE Transactions on Information Theory

Volumn 52, Issue 3, 2006, Pages 1017-1029

  Eriksson, Jan ^a,b   Koivunen, Visa ^a,b  

^a IEEE   (Finland)

^b AALTO UNIVERSITY   (Finland)

Author keywords

Blind methods; Circularity; Complex darmois skitovich theorem; Complex linear models; Differential entropy; Independent component analysis (ICA); Noncircular complex random vectors; Properness

Indexed keywords

BLIND METHODS; CIRCULARITY; COMPLEX DARMOIS-SKITOVICH THEOREM; COMPLEX LINEAR MODELS; DIFFERENTIAL ENTROPY; NONCIRCULAR COMPLEX RANDOM VECTORS; PROPERNESS;

COMPUTATIONAL COMPLEXITY; INDEPENDENT COMPONENT ANALYSIS; MATHEMATICAL MODELS; PROBLEM SOLVING; THEOREM PROVING;

VECTORS;

EID: 33745701263     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2005.864440     Document Type: Article

References (36)

1. P. Comon, "Independent component analysis, a new concept?," Signal Process., vol. 36, no. 3, pp. 287-314, Apr. 1994.
2. A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis. New York: Wiley, 2001.
3. A. Cichocki and S. Amari, Adaptive Blind Signal and Image Processing: Learning Algorithms and Applications. New York: Wiley, 2002.
4. A. Kagan, Y. Linnik, and C. Rao, Characterization Problems in Mathematical Statistics, ser. Probability and Mathematical Statistics. New York: Wiley, 1973.
5. J. Eriksson and V. Koivunen, "Identifiability, separability and uniqueness of linear ICA models," IEEE Signal Process. Lett., vol. 11, no. 7, pp. 601-604, July 2004.
6. J.-F. Cardoso, "An efficient technique for the blind separation of complex sources," in Proc. IEEE Int. Workshop on Higher-Order Statistics (HOS'93), South Lake Tahoe, CA, Jun. 1993, pp. 275-279.
7. E. Bingham and A. Hyvärinen, "A fast fixed-point algorithm for independent component analysis of complex valued signals," Int. J. Neural Syst., vol. 10, no. 1, pp. 1-8, Feb. 2000.
8. V. Calhoun and T. Adali, "Complex infomax: Convergence and approximation of infomax with complex nonlinearities," in Proc. IEEE Int. Workshop on Neural Networks for Signal Processing (NNSP2002), Martigny, Switzerland, Sep. 2002, pp. 307-316.
9. S. Fiori, "Extended hebbian learning for blind separation of complex-valued sources," IEEE Trans. Circuits Syst. II, vol. 50, no. 4, pp. 195-202, Apr. 2003.
10. J. Anemüller, T. Sejnowski, and S. Makeig, "Complex independent component analysis of frequency-domain electroencephalographic data," Neural Netw., vol. 16, no. 9, pp. 1311-1323, Nov. 2003.
11. F. Neeser and J. Massey, "Proper complex random processes with applications to information theory," IEEE Trans. Inf. Theory, vol. 39, no. 4, pp. 1293-1302, Jul. 1993.
12. B. Picinbono, "Second-order complex random vectors and normal distributions," IEEE Trans. Signal Process., vol. 44, no. 10, pp. 2637-2640, Oct. 1996.
13. P. Schreier and L. Scharf, "Second-order analysis of improper complex random vectors and processes," IEEE Trans. Signal Process., vol. 51, no. 3, pp. 714-725, Mar. 2003.
14. N. Vakhania and N. Kandelaki, "Random vectors with values in complex Hilbert spaces," Theory Probab. Appl., vol. 41, no. 1, pp. 116-131, Feb. 1996.
15. B. Picinbono and P. Bondon, "Second-order statistics of complex signals," IEEE Trans. Signal Process., vol. 45, no. 2, pp. 411-420, Feb. 1997.
16. O. Grellier, P. Comon, B. Mourrain, and P. Trebuchet, "Analytical blind channel identification," IEEE Trans. Signal Process., vol. 50, no. 9, pp. 2196-2207, Sep. 2002.
17. N. Vakhania, V. Tarieladze, and S. Chobanyan, Probability Distributions on Banach Spaces. Dordrecht, The Netherlands: Reidel, 1987.
18. I. Ruzsa and G. Székely, Algebraic Propbability Theory. New York: Wiley, 1988.
19. R. Dudley, Real Analysis and Probability. New York: Chapman and Hall, 1989.
20. G. Feldman, Arithmetic of Probability Distributions, and Characterization Problems on Abelian Groups, ser. Translations of Mathematical Monographs. Providence, RI: AMS, 1993, vol. 116.
21. R. Wooding, "The multivariate distribution of complex normal variables," Biometrika, vol. 43, no. 1/2, pp. 212-215, Jun. 1956.
22. N. Goodman, "Statistical analysis based on certain multivariate complex Gaussian distribution (an introduction)," Ann. Math. Statist., vol. 34, no. 1, pp. 152-177, Mar. 1963.
23. P. Amblard, M. Gaeta, and J. Lacoume, "Statistics for complex variables and signals - Part I: Variables," Signal Process., vol. 53, pp. 1-13, 1996.
24. P. Krishnaiah and J. Lin, "Complex elliptically symmetric distributions," Commun. Statist. A, vol. 15, no. 12, pp. 3693-3718, 1986.
25. W. Hudson and J. Veeh, "Complex stable sums of complex stable random variables," J. Mult. Anal., vol. 77, pp. 229-238, 2001.
26. N. Vakhania, "Polya's characterization theorem for complex random variables," J. Complex., vol. 13, pp. 480-488, 1997.
27. R. Horn and C. Johnson, Matrix Analysis. New York: Cambridge Univ. Press, 1985.
28. B. Picinbono and P. Chevalier, "Widely linear estimation with complex data," IEEE Trans. Signal Process., vol. 43, no. 8, pp. 2030-2033, Aug. 1995.
29. R. Cuppens, Decomposition of Multivariate Probabilities, ser. Probability and Mathematical Statistics. New York: Academic, 1975, vol. 29.
30. T. Cover and J. Thomas, Elements of Information Theory. New York: Wiley, 1991.
31. O. Tauböck, "Rotationally variant complex channels," in Proc. 23rd Symp. Information Theory in the Benelux, Louvain-la-Neuve, Belgium, May 2001
32. S. Ghurye and I. Olkin, "A characterization of the multivariate normal distribution," Ann. Math. Statist., vol. 33, no. 2, pp. 533-541, Jun. 1962.
33. A. Mathai and O. Pederzoli, Characterizations of the Normal Probability Law. New Delhi, India: Wiley Eastern Ltd., 1977.
34. Y. Linnik and I. Ostrovskiǐ, Decomposition of Random Variables and Vectors, ser. Translation of Mathematical Monographs: AMS, 1977, vol. 48.
35. J. Aczél and J. Dhombres, Functional Equations in Several Variables, ser. Encyclopedia of Mathematics and its Applications. Cambridge, U.K.: Cambridge Univ. Press, 1989, vol. 31.
36. J. Aczél, Lectures on Functional Equations and Their Applications, ser. Mathematics in Science and Engineering. New York: Academic, 1966, vol. 19.