Abstract
In this paper, we discuss the joint blind source separation (JBSS) of real-valued Gaussian stationary sources with uncorrelated samples from a new perspective. We show that the second-order statistics of the observations can be reformulated as a coupled decomposition of several tensors. The canonical polyadic decomposition (CPD) of each such tensor, if unique, results in the identification of one or two mixing matrices. The proposed new formulation implies that standard algorithms for joint diagonalization and CPD may be used to estimate the mixing matrices, although only in a sub-optimal manner. We discuss the uniqueness and identifiability of this new approach. We demonstrate how the proposed approach can bring new insights on the uniqueness of JBSS in the presence of underdetermined mixtures.
D. Lahat—This work is supported by the project CHESS, 2012-ERC-AdG-320684. GIPSA-Lab is a partner of the LabEx PERSYVAL-Lab (ANR–11-LABX-0025).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In practice, due to finite sample size and noise, (13) is just an approximation. Questions related to uniqueness and estimation in the presence of perturbations from the exact model are beyond the scope of this paper.
References
Kim, T., Eltoft, T., Lee, T.-W.: Independent vector analysis: an extension of ICA to multivariate components. In: Rosca, J., Erdogmus, D., PrÃncipe, J.C., Haykin, S. (eds.) ICA 2006. LNCS, vol. 3889, pp. 165–172. Springer, Heidelberg (2006). https://doi.org/10.1007/11679363_21
Li, Y.O., Adalı, T., Wang, W., Calhoun, V.D.: Joint blind source separation by multiset canonical correlation analysis. IEEE Trans. Signal Process. 57(10), 3918–3929 (2009)
Hitchcock, F.L.: The expression of a tensor or a polyadic as a sum of products. J. Math. Phys. 6(1), 164–189 (1927)
Li, X.L., Adalı, T., Anderson, M.: Joint blind source separation by generalized joint diagonalization of cumulant matrices. Signal Process. 91(10), 2314–2322 (2011)
Congedo, M., Phlypo, R., Chatel-Goldman, J.: Orthogonal and non-orthogonal joint blind source separation in the least-squares sense. In: Proceedings of the EUSIPCO, Bucharest, pp. 1885–1889, August 2012
Lahat, D., Jutten, C.: Joint analysis of multiple datasets by cross-cumulant tensor (block) diagonalization. In: Proceedings of the SAM, Rio de Janeiro, July 2016
Lahat, D., Jutten, C.: Joint independent subspace analysis using second-order statistics. IEEE Trans. Signal Process. 64(18), 4891–4904 (2016)
Kruskal, J.B.: Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics. Proc. London Math. Soc. 18(2), 95–138 (1977)
Lahat, D., Adalı, T., Jutten, C.: Multimodal data fusion: an overview of methods, challenges and prospects. Proc. IEEE 103(9), 1449–1477 (2015)
Anderson, T.W.: An introduction to multivariate statistical analysis. Wiley, New York (1958)
VÃa, J., Anderson, M., Li, X.L., Adalı, T.: A maximum likelihood approach for independent vector analysis of Gaussian data sets. In: Proceedings of the MLSP, Beijing, September 2011
VÃa, J., Anderson, M., Li, X.L., Adalı, T.: Joint blind source separation from second-order statistics: Necessary and sufficient identifiability conditions. In: Proceedings of the ICASSP, Prague, pp. 2520–2523, May 2011
Anderson, M., Fu, G.S., Phlypo, R., Adalı, T.: Independent vector analysis: identification conditions and performance bounds. IEEE Trans. Signal Process. 62(17), 4399–4410 (2014)
Lahat, D., Jutten, C.: An alternative proof for the identifiability of independent vector analysis using second order statistics. In: Proceedings of the ICASSP, Shanghai, March 2016
Gong, X.F., Lin, Q.H., Cong, F.Y., De Lathauwer, L.: Double coupled canonical polyadic decomposition for joint blind source separation (2017). arXiv:1612.09466 [stat.ML]
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Lahat, D., Jutten, C. (2018). A New Link Between Joint Blind Source Separation Using Second Order Statistics and the Canonical Polyadic Decomposition. In: Deville, Y., Gannot, S., Mason, R., Plumbley, M., Ward, D. (eds) Latent Variable Analysis and Signal Separation. LVA/ICA 2018. Lecture Notes in Computer Science(), vol 10891. Springer, Cham. https://doi.org/10.1007/978-3-319-93764-9_17
Download citation
DOI: https://doi.org/10.1007/978-3-319-93764-9_17
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-93763-2
Online ISBN: 978-3-319-93764-9
eBook Packages: Computer ScienceComputer Science (R0)