## Joint blind source separation (JBSS) Approaches

Matlab code for six JBSS algorithms are presented:

- Multiset canonical correlation analysis (MCCA) [1,2]
- Independent vector analysis using multivariate Gaussian distribution prior (IVA-G) [3,4,5]
- Joint diagonalization using second-order statistics (JDIAG-SOS) [6,7]
- Independent vector analysis using second-order uncorrelated multivariate Laplace distribution prior (IVA-L) [8,9,10] and the decoupled version (IVA-L-Decp) [4]
- Joint diagonalization using fourth-order cumulant (JDIAG-CUM4) [6,7]
- Independent vector analysis using multivariate power exponential distribution prior (IVA-MPE) also known as (IVA-GGD) [11]

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.