Complex likelihood maximization

T. Adali, H. Li, M. Novey, and J.-F. Cardoso, "Complex ICA using nonlinear functions," IEEE Trans. Signal Processing, vol. 56, no. 9, pp. 4536 -4544, Sept. 2008.

Brief introduction:
This algorithm is derived using maximization of likelihood principle. The form of the update rule can be treated as a direct extension of real-valued infomax algorithm. The score function used in the update rule may take different forms, such as 'atan' or 'asinh', which corresponding to different density models. The algorithm can be implemented with or without a unitary constraint on the separating matrix.

Parameters of the algorithm:
X: mixtures
Type: to choose different score functions

Nonlinear decorrelations

T. Adali, T. Kim, and V. Calhoun, "Independent component analysis by complex nonlinearities," in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing (ICASSP), Montreal, Canada, May 2004, vol. 5, pp. 525-528.

Brief introduction:
This algorithm can be treated as an extension of real-valued nonlinear decorrelation ICA algorithm to the complex domain. This algorithm is based on the definition of independent random variables, i.e., independence of two random variables implies that the two random variables are nonlinear decorrelated for any nonlinear functions. This principle yields an update rule similar to the complex ML ICA update rule.

Parameters of the algorithm:
X: mixtures
Type: to choose the nonlinear function used in the update rule

Complex ICA-EBM

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.

Brief introduction:
To derive the ICA-EBM algorithm, the authors first introduce a new differential entropy estimator for complex random variables by approximating the entropy estimate using a numerically computed maximum entropy bound. The associated maximum entropy distributions belong to the class of weighted linear combinations and elliptical distributions, and together, they provide a rich array of bivariate distributions for density matching. Then the ICA-EBM algorithm is derived using the new entropy estimator and a line search optimization procedure. Complex ICA-EBM can efficiently exploit both the noncircularity and non-Gaussianity of sources for separation. Simulation results demonstrated the superior separation performance and computational efficiency.

Parameters of the algorithm:
X: mixtures