## Independent component analysis (ICA) by entropy bound minimization (EBM) and entropy rate minimization (ERM)

Matlab code for five ICA algorithms are presented:

- Real-valued ICA by entropy bound minimization (ICA-EBM) [1,2]
- Real-valued ICA by entropy rate bound minimization (ICA-ERBM) [3]
- Real-valued ICA by entropy rate minimization (ICA-ERM) [4]
- Complex-valued ICA by entropy bound minimization (complex ICA-EBM) [5]
- Complex-valued ICA by entropy rate bound minimization (complex ICA-ERBM) [6]

### Real-valued ICA-EBM

ICA by entropy bound minimization provides flexible density matching through use of four measuring functions based on the maximum entropy principle. Four nonlinearities are used as measuring functions for calculating the entropy bound, and the associated maximum entropy density can be symmetric or skewed, heavy-tailed or not heavy-tailed.

### Real-valued ICA-ERBM

ICA by entropy rate bound minimization takes both non-Gaussianity and sample correlation into account by minimizing mutual information rate. It is originally introduced as Full Blind Source Separation (FBSS). The algorithm By assuming the sources are outputs of linear systems driven by independently and identically distributed (i.i.d.) noise, the entropy rate estimation problem is converted to an entropy estimation problem solved using EBM.

### Real-valued ICA-ERM

ICA by entropy rate minimization (ERM) methods use the mutual information rate, which leads to the minimization of entropy rate, as the cost function to take both non-Gaussianity and sample dependence into account. The estimation of entropy rate is the most difficult part of the problem. The ERM methods estimate entropy rate by assuming Markovian or invertible filter source model. Entropy rate minimization via multivariate generalized Gaussian distribution (ERM-MG) assumes Markovian model with multivariate generalized Gaussian distribution as the source prior. Both entropy rate minimization via AR driven by GGD process (ERM-ARG) and entropy rate bound minimization via AR source (ERBM-AR) assume the source is generated by an AR model driven by an i.i.d.~process. For ERM-ARG, the innovation process is modeled by a generalized Gaussian distribution. For ERBM-AR, the distribution of the innovation process is assumed to be unknown and modeled by a maximum entropy distribution.

### Complex ICA-EBM

This is the complex version of the EBM algorithm. Unlike real-valued ICA where a univariate density is approximated, here a bivariate density model is used. In complex ICA-EBM, weighted linear combinations and elliptical distributions are considered, and they provide a rich array of bivariate distributions for density matching by starting again the four measuring functions used for real-valued EBM.

### Complex ICA-ERBM

CERBM is the complex version of the ERBM algorithm. It uses the mutual information rate as the cost function, and estimates the entropy rate by assuming signal is generated by an invertible filter driven by an i.i.d. random process. CERBM exploits all three important types of diversity: non-Gaussianity, sample dependence, and noncircularity.

**References:**

[1] X.-L. Li and T. Adali, "A novel entropy estimator and its application to ICA," in Proc. IEEE Workshop on Machine Learning for Signal Processing (MLSP), Grenoble, France, Sep. 2009.

[2] X.-L. Li and T. Adali, "Independent component analysis by entropy bound minimization,"

*IEEE Trans. Signal Processing,*vol. 58, no. 10, pp. 5151-5164, Oct. 2010.

[3] X.-L. Li, and T. Adali, "Blind spatiotemporal separation of second and/or higher-order correlated sources by entropy rate minimization," in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing (ICASSP), Dallas, TX, March 2010.

[4] G.-S. Fu, R. Phlypo, M. Anderson, X.-L. Li, and T. Adali,"Blind source separation by entropy rate minimization,"

*IEEE Trans. Signal Processing,*vol. 62, no. 16, pp. 4245-4255, Aug. 2014.

[5] X.-L. Li and T. Adali, "Complex independent component analysis by entropy bound minimization,"

*IEEE Trans. Circuits and Systems I,*vol. 57, no. 7, pp. 1417-1430, July 2010.

[6] G.-S. Fu, R. Phlypo, M. Anderson, and T. Adali, "Complex Independent Component Analysis Using Three Types of Diversity: Non-Gaussianity, Nonwhiteness, and Noncircularity,"

*IEEE Trans. Signal Processing,*vol. 63, no. 3, pp. 794-805, Feb. 2015.