Joint blind source separation (JBSS) Approaches

Matlab code for six JBSS algorithms are presented:

  1. Multiset canonical correlation analysis (MCCA) [1,2]
  2. Independent vector analysis using multivariate Gaussian distribution prior (IVA-G) [3,4,5]
  3. Joint diagonalization using second-order statistics (JDIAG-SOS) [6,7]
  4. Independent vector analysis using second-order uncorrelated multivariate Laplace distribution prior (IVA-L) [8,9,10] and the decoupled version (IVA-L-Decp) [4]
  5. Joint diagonalization using fourth-order cumulant (JDIAG-CUM4) [6,7]
  6. Independent vector analysis using multivariate power exponential distribution prior (IVA-MPE) also known as (IVA-GGD) [11]
These algorithms are different solutions for JBSS (or alternatively solutions to MCCA). JBSS algorithms attempt to achieve BSS on multiple datasets simultaneously by ultimately balancing two criterion, maximization of the estimated source independence within a dataset and maximization of source dependence across datasets. JBSS can be formulated as a very special case of multidimensional independent component analysis (MICA).

A variety of orthogonal and nonorthogonal approaches based on second-order and higher-order statistics using maximum likelihood (or equivalently mutual information minimization) and joint diagonalization have been used to perform both real and complex-valued JBSS.

MCCA

This is a group of algorithms that use cost functions based on second-order statistics. The implementation estimates an orthogonal matrix using a deflationary approach. It ignores sample-to-sample dependence and higher-order statistics.

IVA-G

This is an algorithm that uses second-order statistics and a decoupling trick to estimate nonorthogonal demixing matrices by minimizing mutual information assuming a multivariate Gaussian prior. If the data is complex-valued then the pseudo-covariance matrix is included in the mutual information measure. It ignores sample-to-sample dependence and higher-order statistics.

JDIAG-SOS

This is an algorithm that uses symmetric orthogonal joint diagonalization of covariance matrices based on multiple datasets. It is the only algorithm that explicitly can exploit sample-to-sample dependence to improve source separation. It ignores higher-order statistics.

IVA-L

This is an algorithm that uses higher-order statistics to estimate nonorthogonal demixing matrices by minimizing mutual information assuming a second-order uncorrelated multivariate Laplace prior. It ignores sample-to-sample dependence and second-order statistics. IVA-L is implemented using the a matrix relative gradient and IVA-L-Decp using a vector gradient via a decoupling method.

JDIAG-CUM4

This is an algorithm that uses symmetric orthogonal joint diagonalization of fourth-order cumulants based. It ignores sample-to-sample dependence and second-order statistics.

IVA-GGD

This is an algorithm that uses all order statistics to estimate non-orthogonal demixing matrices by minimizing mutual information assuming a multivariate generalized Gaussian prior. It ignores sample-to-sample dependence. This method was initially introduced as IVA with a multivariate power exponential prior (IVA-MPE).
References:

[1] J. R. Kettenring, "Canonical analysis of several sets of variables Biometrika," 1971, 58, 433-451
[2] Y.-O. Li, T. Adali, W. Wang, V. D. Calhoun, "Joint Blind Source Separation by Multiset Canonical Correlation Analysis," IEEE Trans. Signal Process., 2009, 57, 3918-3929
[3] M. Anderson, X.-L. Li, & T. Adali, "Nonorthogonal Independent Vector Analysis Using Multivariate Gaussian Model," LNCS: Independent Component Analysis and Blind Signal Separation, Latent Variable Analysis and Signal Separation, Springer Berlin / Heidelberg, 2010, 6365, 354-361
[4] M. Anderson, T. Adali, & X.-L. Li, "Joint Blind Source Separation of Multivariate Gaussian Sources: Algorithms and Performance Analysis," IEEE Trans. Signal Process., 2012, 60, 1672-1683
[5] M. Anderson, X.-L. Li, & T. Adali, "Complex-valued Independent Vector Analysis: Application to Multivariate Gaussian Model," Signal Process., 2012, 1821-1831
[6] X.-L. Li, T. Adali, & M. Anderson, "Joint Blind Source Separation by Generalized Joint Diagonalization of Cumulant Matrices," Signal Process., 2011, 91, 2314-2322
[7] X.-L. Li, M. Anderson, & T. Adali, "Second and Higher-Order Correlation Analysis of Multiset Multidimensional Variables by Joint Diagonalization," Lecture Notes in Computer Science: Independent Component Analysis and Blind Signal Separation, Latent Variable Analysis and Signal Separation, Springer Berlin / Heidelberg, 2010, 6365, 197-204
[8] T. Kim, I. Lee, & T.-W. Lee, "Independent Vector Analysis: Definition and Algorithms," Proc. of 40th Asilomar Conference on Signals, Systems, and Computers, 2006, 1393-1396
[9] T. Kim, T. Eltoft, & T.-W. Lee, "Independent Vector Analysis: an extension of ICA to multivariate components," Lecture Notes in Computer Science: Independent Component Analysis and Blind Signal Separation, Independent Component Analysis and Blind Signal Separation, Springer Berlin / Heidelberg, 2006, 3889, 165-172
[10] T. Kim, H. T. Attias, S.-Y. Lee, & T.-W. Lee, "Blind Source Separation Exploiting Higher-Order Frequency Dependencies," IEEE Trans. Audio Speech Lang. Process., 2007, 15, 70-79
[11] M. Anderson, G.-S. Fu, R. Phlypo, and T. Adali, "Independent Vector Analysis: Identification Conditions and Performance Bounds," IEEE Trans. Signal Processing, vol. 62, no. 17, pp. 4399--4410, Sep. 2014.