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A Novel Underdetermined Source Recovery Algorithm Based on k-Sparse Component Analysis

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Abstract

Sparse component analysis (SCA) is a popular method for addressing underdetermined blind source separation in array signal processing applications. We are motivated by problems that arise in the applications where the sources are densely sparse (i.e. the number of active sources is high and very close to the number of sensors). The separation performance of current underdetermined source recovery (USR) solutions, including the relaxation and greedy families, reduces with decreasing the mixing system dimension and increasing the sparsity level (k). In this paper, we present a k-SCA-based algorithm that is suitable for USR in low-dimensional mixing systems. Assuming the sources is at most \((m-1\)) sparse where m is the number of mixtures; the proposed method is capable of recovering the sources from the mixtures given the mixing matrix using a subspace detection framework. Simulation results show that the proposed algorithm achieves better separation performance in k-SCA conditions compared to state-of-the-art USR algorithms such as basis pursuit, minimizing norm-L1, smoothed L0, focal underdetermined system solver and orthogonal matching pursuit.

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Notes

  1. http://ee.sharif.edu/~SLzero/.

  2. http://www.cs.ubc.ca/~mpf/spgl1/.

  3. https://www.mathworks.com/matlabcentral/fileexchange/48641-sparse-blind-source-separation-sparse-component-analysis-modal-identification.

  4. http://dsp.ucsd.edu/jfmurray/software.htm and https://sites.google.com/site/researchbyzhang/softwareWe tested both available codes. Their performances were similar, but the first code was very slower than second one. Therefore, we just reported the results of second code.

  5. https://www.mathworks.com/matlabcentral/fileexchange/32402-cosamp-and-omp-for-sparse-recovery.

References

  1. F. Abrard, Y. Deville, Blind separation of dependent sources using the “time–frequency ratio of mixtures” approach, in Seventh International Symposium on Signal Processing and Its Applications. Proceedings, vol. 2 (IEEE, 2003), pp. 81–84

  2. F. Abrard, Y. Deville, A time-frequency blind signal separation method applicable to underdetermined mixtures of dependent sources. Signal Process. 85(7), 1389–1403 (2005)

    Article  MATH  Google Scholar 

  3. A. Aissa-El-Bey, N. Linh-Trung, K. Abed-Meraim, A. Belouchrani, Y. Grenier, Underdetermined blind separation of nondisjoint sources in the time–frequency domain. IEEE Trans. Signal Process. 55(3), 897–907 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. S. Araki, S. Makino, A. Blin, R. Mukai, H. Sawada, Underdetermined blind separation for speech in real environments with sparseness and ICA, in IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings. (ICASSP’04), vol. 3 (IEEE, 2004), pp. iii–881

  5. S. Araki, H. Sawada, R. Mukai, S. Makino, Underdetermined blind sparse source separation for arbitrarily arranged multiple sensors. Signal Process. 87(8), 1833–1847 (2007)

    Article  MATH  Google Scholar 

  6. P. Bofill, M. Zibulevsky, Underdetermined blind source separation using sparse representations. Signal process. 81(11), 2353–2362 (2001)

    Article  MATH  Google Scholar 

  7. S.S. Chen, D.L. Donoho, M.A. Saunders, Atomic decomposition by basis pursuit. SIAM Rev. 43(1), 129–159 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Chichocki, Y. Li, P. Georgiev, S Amari, Beyond ICA: robust sparse signal representations, in Proceedings of the 2004 International Symposium on Circuits and Systems, ISCAS’04, vol. 5 (IEEE, 2004), pp. V–V

  9. A. Cichocki, S. Amari, Adaptive blind signal and image processing: learning algorithms and applications, vol 1. (John Wiley & Sons, 2002)

  10. P. Comon, C. Jutten, Handbook of Blind Source Separation: Independent Component Analysis and Applications (Academic press, Cambridge, 2010)

    Google Scholar 

  11. D. Coppersmith, S. Winograd, Matrix multiplication via arithmetic progressions. J. Symb. Comput. 9(3), 251–280 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  12. S.F. Cotter, B.D. Rao, K. Engan, K. Kreutz-Delgado, Sparse solutions to linear inverse problems with multiple measurement vectors. IEEE Trans. Signal Process. 53(7), 2477–2488 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. D.L. Donoho, M. Elad, V.N. Temlyakov, Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Trans. Inf. Theory 52(1), 6–18 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing, 1st edn. (Springer Publishing Company, Incorporated, New York, 2010)

    Book  MATH  Google Scholar 

  15. E. Eqlimi, B. Makkiabadi, An efficient K-SCA based unerdetermined channel identification algorithm for online applications, in 23rd European Signal Processing Conference (EUSIPCO) (IEEE, 2015), pp. 2661–2665

  16. E. Eqlimi, B. Makkiabadi, Multiple sparse component analysis based on subspace selective search algorithm, in 2015 23rd Iranian Conference on Electrical Engineering (ICEE) (IEEE, 2015), pp. 550–554

  17. N.D. Freitas, Y. Wang, M. Mahdaviani, D. Lang, in Fast Krylov methods for n-body learning, ed. by Y. Weiss, B. Schölkopf, J.C. Platt. Advances in Neural Information Processing Systems (MIT Press, 2006), pp. 251–258

  18. P. Georgiev, F. Theis, A. Cichocki, Blind source separation and sparse component analysis of overcomplete mixtures, in IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings. (ICASSP’04), vol. 5 (IEEE, 2004), pp. V–493

  19. P. Georgiev, F. Theis, A. Cichocki, Sparse component analysis and blind source separation of underdetermined mixtures. IEEE Trans. Neural Netw. 16(4), 992–996 (2005). https://doi.org/10.1109/TNN.2005.849840

    Article  Google Scholar 

  20. P. Georgiev, F. Theis, A. Cichocki, H. Bakardjian, in Sparse component analysis: a new tool for data mining, ed. by P.M. Pardalos, V.L. Boginski, A. Vazacopoulos. Data Mining in Biomedicine. Springer Optimization and Its Applications, vol. 7 (Springer, Boston, MA, 2007), pp. 91–116

  21. I.F. Gorodnitsky, B.D. Rao, Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm. IEEE Trans. Signal Process. 45(3), 600–616 (1997)

    Article  Google Scholar 

  22. R. Gribonval, M. Nielsen, Sparse representations in unions of bases. IEEE Trans. Inf. Theory 49(12), 3320–3325 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. R. Gribonval, P. Vandergheynst, On the exponential convergence of matching pursuits in quasi-incoherent dictionaries. IEEE Trans. Inf. Theory 52(1), 255–261 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. B. Hassibi, An efficient square-root algorithm for blast, in 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing. ICASSP’00. Proceedings, vol. 2 (IEEE, 2000), pp. II737–II740

  25. Z. He, A. Cichocki, Y. Li, S. Xie, S. Sanei, K-hyperline clustering learning for sparse component analysis. Signal Process. 89(6), 1011–1022 (2009)

    Article  MATH  Google Scholar 

  26. A. Hyvärinen, J. Karhunen, E. Oja, Independent Component Analysis, vol. 46 (Wiley, Hoboken, 2004)

    Google Scholar 

  27. A. Jourjine, S. Rickard, O. Yilmaz, Blind separation of disjoint orthogonal signals: Demixing n sources from 2 mixtures, in 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing. ICASSP’00. Proceedings, vol. 5 (IEEE, 2000), pp. 2985–2988

  28. F. Le Gall, Powers of tensors and fast matrix multiplication, in Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation (ACM, 2014), pp. 296–303

  29. Y. Li, S.I. Amari, A. Cichocki, D.W. Ho, S. Xie, Underdetermined blind source separation based on sparse representation. IEEE Trans. Signal Process. 54(2), 423–437 (2006)

    Article  MATH  Google Scholar 

  30. Y. Li, A. Cichocki, S.I. Amari, Sparse component analysis for blind source separation with less sensors than sources, in ICA2003 (Citeseer, 2003), pp. 89–94

  31. Y. Li, A. Cichocki, S.I. Amari, Blind estimation of channel parameters and source components for EEG signals: a sparse factorization approach. IEEE Trans. Neural Netw. 17(2), 419–431 (2006)

    Article  Google Scholar 

  32. Y. Li, Z.L. Yu, N. Bi, Y. Xu, Z. Gu, S. Amari, Sparse representation for brain signal processing: a tutorial on methods and applications. IEEE Signal Process. Mag. 31(3), 96–106 (2014)

    Article  Google Scholar 

  33. B. Liu, V.G. Reju, A.W. Khong, A linear source recovery method for underdetermined mixtures of uncorrelated AR-model signals without sparseness. IEEE Trans. Signal Process. 62(19), 4947–4958 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  34. S.G. Mallat, Z. Zhang, Matching pursuits with time–frequency dictionaries. IEEE Trans. Signal Process. 41(12), 3397–3415 (1993)

    Article  MATH  Google Scholar 

  35. F. Marvasti, A. Amini, F. Haddadi, M. Soltanolkotabi, B.H. Khalaj, A. Aldroubi, S. Sanei, J. Chambers, A unified approach to sparse signal processing. EURASIP J. Adv. Signal Process. 2012(1), 44 (2012)

    Article  Google Scholar 

  36. R. Mises, H. Pollaczek-Geiringer, Praktische verfahren der gleichungsauflösung. ZAMM J. Appl. Math. Mech. (Zeitschrift für Angewandte Mathematik und Mechanik) 9(1), 58–77 (1929)

    Article  MATH  Google Scholar 

  37. G.H. Mohimani, M. Babaie-Zadeh, C. Jutten, Fast sparse representation based on smoothed 0 norm, in International Conference on Independent Component Analysis and Signal Separation (Springer, 2007), pp. 389–396

  38. H. Mohimani, M. Babaie-Zadeh, C. Jutten, A fast approach for overcomplete sparse decomposition based on smoothed norm. IEEE Trans. Signal Process. 57(1), 289–301 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. D. Needell, J.A. Tropp, Cosamp: iterative signal recovery from incomplete and inaccurate samples. Appl. Comput. Harmon. Anal. 26(3), 301–321 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  40. M. Niknazar, H. Becker, B. Rivet, C. Jutten, P. Comon, Blind source separation of underdetermined mixtures of event-related sources. Signal Process. 101, 52–64 (2014)

    Article  Google Scholar 

  41. D. Peng, Y. Xiang, Underdetermined blind source separation based on relaxed sparsity condition of sources. IEEE Trans. Signal Process. 57(2), 809–814 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  42. Y. Saad, Krylov subspace methods for solving large unsymmetric linear systems. Math. Comput. 37(155), 105–126 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  43. I. Takigawa, M. Kudo, J. Toyama, Performance analysis of minimum 1-norm solutions for underdetermined source separation. IEEE Trans. Signal Process. 52(3), 582–591 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  44. F.J. Theis, P.G. Georgiev, A. Cichocki, Robust overcomplete matrix recovery for sparse sources using a generalized Hough transform, in ESANN (Citeseer, 2004), pp. 343–348

  45. J.A. Tropp, Greed is good: algorithmic results for sparse approximation. IEEE Trans. Inf. Theory 50(10), 2231–2242 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  46. E. Van Den Berg, M.P. Friedlander, Probing the Pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput. 31(2), 890–912 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  47. E. Van den Berg, M.P. Friedlander, Spgl1: a solver for large-scale sparse reconstruction (2007). http://www.cs.ubc.ca/labs/scl/spgl1

  48. Y. Washizawa, A. Cichocki, On-line k-plane clustering learning algorithm for sparse component analysis, in Acoustics, Speech and Signal Processing. ICASSP 2006 Proceedings, vol. 5 (IEEE, 2006), pp. V–V

  49. J. Wen, H. Liu, S. Zhang, M. Xiao, A new fuzzy K-EVD orthogonal complement space clustering method. Neural Comput. Appl. 24(1), 147–154 (2014)

    Article  Google Scholar 

  50. Y. Wen, Z. Hongyi, Blind source separation based on K-SCA assumption, in 2010 3rd IEEE International Conference on Computer Science and Information Technology (ICCSIT), vol. 9 (2010), pp. 116–121. https://doi.org/10.1109/ICCSIT.2010.5564818

  51. S. Winter, H. Sawada, S. Araki, S. Makino, Overcomplete BSS for convolutive mixtures based on hierarchical clustering, in International Conference on Independent Component Analysis and Signal Separation (Springer, 2004), pp. 652–660

  52. D. Wubben, R. Bohnke, V. Kuhn, K.D. Kammeyer, MMSE extension of V-blast based on sorted QR decomposition, in 2003 IEEE 58th Vehicular Technology Conference. VTC 2003-Fall, vol. 1 (IEEE, 2003), pp. 508–512

  53. C.J. Yao, H.L. Liu, Z.T. Cui, Mixing matrix recovery of underdetermined source separation based on sparse representation, in 2007 International Conference on Computational Intelligence and Security (IEEE, 2007), pp. 1–5

  54. H. Zayyani, M. Babaie-Zadeh, C. Jutten, An iterative Bayesian algorithm for sparse component analysis in presence of noise. IEEE Trans. Signal Process. 57(11), 4378–4390 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  55. M. Zibulevsky, B.A. Pearlmutter, Blind source separation by sparse decomposition in a signal dictionary. Neural comput. 13(4), 863–882 (2001)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Medical Physics and Biomedical Engineering, School of Medicine, Tehran University of Medical Sciences (TUMS), Tehran, Iran

    Ehsan Eqlimi, Bahador Makkiabadi, Nasser Samadzadehaghdam & Hassan Khajehpour

  2. WAVES Research Group, Department of Information Technology (INTEC), Ghent University (UGent), Ghent, Belgium

    Ehsan Eqlimi

  3. Research Center for Biomedical Technology and Robotics (RCBTR), Institute of Advanced Medical Technologies (IAMT), TUMS, Tehran, Iran

    Ehsan Eqlimi, Bahador Makkiabadi, Nasser Samadzadehaghdam & Hassan Khajehpour

  4. Department of Medical Physics and Biomedical Engineering, School of Medicine, Shahid Beheshti University of Medical Sciences (SBMU), Tehran, Iran

    Fahimeh Mohagheghian

  5. School of Science and Technology, Nottingham Trent University (NTU), Nottingham, UK

    Saeid Sanei

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  1. Ehsan Eqlimi
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  2. Bahador Makkiabadi
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  3. Nasser Samadzadehaghdam
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  4. Hassan Khajehpour
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  5. Fahimeh Mohagheghian
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Correspondence to Bahador Makkiabadi.

Additional information

This study is part of a Ph.D. thesis supported by Tehran University of Medical Sciences (TUMS), Tehran, Iran; Grant No.: 94-01-30-28327. This study is also supported by Cognitive Sciences and Technologies Council (CSTC), Tehran, Iran under tracking code 4907. MATLAB codes of our proposed algorithm can be found for benchmarking and further research: (https://github.com/EhsanEqlimi).

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Eqlimi, E., Makkiabadi, B., Samadzadehaghdam, N. et al. A Novel Underdetermined Source Recovery Algorithm Based on k-Sparse Component Analysis. Circuits Syst Signal Process 38, 1264–1286 (2019). https://doi.org/10.1007/s00034-018-0910-9

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