Fourth-order cumulant-based blind identification of underdetermined mixtures

IEEE Transactions on Signal Processing

Volumn 55, Issue 6 II, 2007, Pages 2965-2973

  De Lathauwer, Lieven ^a,b   Castaing, Joséphine ^a   Cardoso, Jean Fraņcois ^c  

^a Centre national de la recherche scientifique   (France)

^b UNIVERSITY OF LEUVEN   (Belgium)

^c TELECOM PARISTECH   (France)

Author keywords

Cumulant; Higher order statistics; Higher order tensor; Independent component analysis (ICA); Parallel factor analysis; Simultaneous diagonalization; Underdetermined mixture

Indexed keywords

COMPUTER SIMULATION; ESTIMATION; INDEPENDENT COMPONENT ANALYSIS; MATRIX ALGEBRA; TENSORS;

MATRIX DIAGONALIZATION; PARALLEL FACTOR ANALYSIS;

BLIND EQUALIZATION;

EID: 34249802620     PISSN: 1053587X     EISSN: None     Source Type: Journal    
DOI: 10.1109/TSP.2007.893943     Document Type: Article

References (42)

1. F. Abrard and Y. Deville, "A time-frequency blind signal separation method applicable to underdetermined mixtures of dependent sources," Signal Process., vol. 85, no. 7, pp. 1389-1403, Jul. 2005.
2. L. Albera, A. Ferréol, P. Comon, and P. Chevalier, "Sixth order blind identification of underdetermined mixtures (BIRTH) of sources," in Proc. 4th Int. Symp. Independent Component Analysis and Blind Signal Separation (ICA'03), Nara, Japan, Apr. 2003.
3. L. Albera, A. Ferréol, P. Comon, and P. Chevalier, "Blind identification of overcomplete mixtures of sources (BIOME)," Lin. Alg. Appl., vol. 391, pp. 1-30, Nov. 2004.
4. A. Belouchrani, K. Abed-Meraim, M. G. Amin, and A. M. Zoubir, "Joint antidiagonalization for blind source separation," in Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing (ICASSP-01), Salt Lake City, UT, May 2001, vol. 5, pp. 2789-2792.
5. P. Bofill and M. Zibulevsky, "Underdetermined blind source separation using sparse representations," Signal Process., vol. 81, no. 11, pp. 2353-2362, Nov. 2001.
6. J.-F. Cardoso, "Super-symmetric decomposition of the fourth-order cumutant tensor. Blind identification of more sources than sensors," in Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing (ICASSP-91), Toronto, ON, Canada, 1991, pp. 3109-3112.
7. J.-F. Cardoso and A. Souloumiac, "Blind beamforming for non-Gaussian signals," Proc. Inst. Elect. Eng., F, vol. 140, no. 6, pp. 362-370, Dec. 1993.
8. J.-F. Cardoso and A. Souloumiac, "Jacobi angles for simultaneous diagonalization," SIAM J. Matrix Anal. Appl., vol. 17, no. 1, pp. 161-164, Jan. 1996.
9. J. Carroll and J. Chang, "Analysis of individual differences in multidimensional scaling via an N-way generalization of 'Eckart-Young' decomposition," Psychometrika, vol. 35, pp. 283-319, 1970.
10. S. S. Chen, D. L. Donoho, and M. A. Saunders, "Atomic decomposition by basis pursuit," Dept. Statist., Stanford Univ., Stanford, CA, Tech. Rep. 30401, 1996.
11. P. Chevalier and A. Ferréol, "On the virtual array concept for the fourth-order direction finding problem," IEEE Trans. Signal Process. vol. 47, no. 9, pp. 2592-2595, Sep. 1999.
12. P. Chevalier, L. Albera, A. Ferréol, and P. Comon, "On the virtual array concept for higher order array processing," IEEE Trans. Signal Process., vol. 53, no. 4, pp. 1254-1271, Apr. 2005.
13. P. Comon, "Independent component analysis, A new concept?," Signal Process., vol. 36, no. 3, pp. 287-314, Apr. 1994.
14. P. Comon and B. Mourrain, "Decomposition of quantics in sums of powers of linear forms," Signal Process., vol. 53, no. 2, pp. 93-107, Sep. 1996.
15. P. Comon, "Blind identification and source separation in 2 × 3 under-determined mixtures," IEEE Trans. Signal Process., vol. 52, no. 1, pp. 11-22, Jan. 2004.
16. P. Comon, "Canonical tensor decompositions," Lab. I3S, Sophia-Antipolis, France, Tech. Rep. RR-2004-17, Jun. 2004.
17. P. Comon and M. Rajih, "Blind identification of under-determined mixtures based on the characteristic function," in Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing (ICASSP'05), Philadelphia, PA, Mar. 18-23, 2005, vol. 4, pp. 1005-1008.
18. L. De Lathauwer, P. Comon, and B. De Moor, "ICA algorithms for 3 sources and 2 sensors," in Proc. 6th Signal Processing Workshop on Higher Order Statistics, Caesarea, Israel, Jun. 14-16, 1999, pp. 116-120.
19. L. De Lathauwer, B. De Moor, and J. Vandewalle, "An algebraic ICA algorithm for 3 sources and 2 sensors," in Proc. 10th Eur. Signal Processing Conf. (EUSIPCO 2000), Tampere, Finland, Sep. 5-8, 2000.
20. L. De Lathauwer, B. De Moor, and J. Vandewalle, "Computation of the canonical decomposition by means of a simultaneous generalized Schur decomposition," SIAM J. Matrix Anal. Appl., vol. 26, pp. 295-327, 2004.
21. L. De Lathauwer, "A link between the canonical decomposition in multilinear algebra and simultaneous matrix diagonalization," SIAM J. Matrix Anal. Appl., vol. 28, no. 3, pp. 642-666, 2006.
22. L. De Lathauwer and J. Castaing, "Second-order blind identification of underdetermined mixtures," in Independent Component Analysis and Blind Source Separation, Proc. 6th Int. Conf. Independent Component Analysis (ICA 2006). Berlin, Germany: Springer-Verlag, Mar. 5-8, 2006, vol. 3889, Lecture Notes in Computer Science, pp. 40-47.
23. L. De Lathauwer and J. Castaing, "Independent component analysis based on simultaneous matrix diagonalization: The underdetermined case," Lab. ETIS, Cergy-Pontoise, France, Tech. Rep., submitted for publication to IEEE Trans. Signal Process.
24. Y. Deville, M. Benali, and F. Abrard, "Differential source separation for underdetermined instantaneous or convolutive mixtures: Concept and algorithms," Signal Process., vol. 84, no. 10, pp. 1759-1776, Oct. 2004.
25. M. C. Dogan and J. M. Mendel, "Applications of cumulants to array processing - Part I: Aperture extension and array calibration," IEEE Trans. Signal Process., vol. 43, no. 5, pp. 1200-1216, May 1995.
26. D. Erdogmus, L. Vielva, and J. C. Principe, "Nonparametric estimation and tracking of the mixing matrix for underdetermined blind source separation," in Proc. ICA'01, San Diego, CA, Dec. 2001, pp. 189-194.
27. J. Eriksson and V. Koivunen, "Identifiability, separability, and uniqueness of linear ICA models," IEEE Signal Process. Lett., vol. 11, no. 7, pp. 601-604, Jul. 2004.
28. A. Ferréol, L. Albera, and P. Chevalier, "Fourth order blind identification of underdetermined mixtures of sources (FOBIUM)," IEEE Trans. Signal Process., vol. 53, no. 5, pp. 1640-1653, May 2005.
29. P. Georgiev, F. Theis, and A. Cichocki, "Sparse component analysis and blind source separation of underdetermined mixtures," IEEE Trans. Neural Netw., vol. 16, no. 5, pp. 992-996, Jul. 2005.
30. R. A. Harshman, "Foundations of the PARAFAC procedure: Model and conditions for an 'explanatory' multi-mode factor analysis," UCLA Working Papers in Phonetics, vol. 16, pp. 1-84, 1970.
31. A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis. New York: Wiley, 2001.
32. J. B. Kruskal, "Rank, decomposition, and uniqueness for 3-way and N way arrays," in Multiway Data Analysis, R. Coppi and S. Bolasco, Eds. Amsterdam, The Netherlands: North-Holland: Elsevier Science, 1989.
33. Y. Li, A. Cichocki, and S.-I. Amari, "Analysis of sparse representation and blind source separation," Neural Comput., vol. 16, no. 6, pp. 1193-1234, Jun. 2004.
34. M. Lewicki and T. J. Sejnowski, "Learning overcomplete representations," Neural Comput., vol. 12, no. 2, pp. 337-365, Feb. 2000.
35. C. R. Rao and S. Mitra, Generalized Inverse of Matrices and Its Applications. New York: Wiley, 1971.
36. N. Sidiropoulos and R. Bro, "On the uniqueness of multilinear decomposition of N-way arrays," J. Chemometrics, vol. 14, no. 3, pp. 229-239, May/Jun. 2000.
37. A. Stegeman, J. M. F. ten Berge, and L. De Lathauwer, "Sufficient conditions for uniqueness in candecomp/parafac and indscal with random component matrices," Psychometrika, vol. 71, no. 2, pp. 219-229, Jun. 2006.
38. A. Taleb, "An algorithm for the blind identification of N independent signals with 2 sensors," in Proc. 16th Int. Symp. on Signal Processing and Its Applications (ISSPA'01), Kuala-Lumpur, Malaysia, Aug. 13-16, 2001, pp. 5-8.
39. F. Theis, E. W. Lang, and C. G. Puntonet, "A geometric algorithm for overcomplete linear ICA," Neurocomputing, vol. 56, pp. 381-398, Jan. 2004.
40. A.-J. van der Veen and A. Paulraj, "An analytical constant modulus algorithm," IEEE Trans. Signal Process., vol. 44, no. 5, pp. 1136-1155, May 1996.
41. A.-J. van der Veen, "A Schur method for low-rank matrix approximation," SIAM J. Matrix Anal. Appl., vol. 17, pp. 139-160, Jan. 1996.
42. A. Yeredor, "Non-orthogonal joint diagonalization in the least-squares sense with application in blind source separation," IEEE Trans. Signal Process., vol. 50, no. 7, pp. 1545-1553, Jul. 2002.