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The present manuscript aims at illustrating fundamental challenges and solutions arising in the design of learning theories by optimization on manifolds in the context of complex-valued neural systems. The special case of a unitary unimodular group of matrices is dealt with. The unitary unimodular group under analysis is a low dimensional and easy-to-handle matrix group. Notwithstanding, it exhibits a rich geometrical structure and gives rise to interesting speculations about methods to solve optimization problems on manifolds. Also, its low dimension allows us to treat most of the quantities involved in computation in closed form as well as to render them in graphical format. Some numerical experiments are presented and discussed within the paper, which deal with complex-valued independent component analysis. [ABSTRACT FROM AUTHOR]

This paper introduces an Independent Component Analysis (ICA) approach to the separation of nonlinear mixtures in the complex domain. Source separation is performed by a complex INFOMAX approach. The neural network which realizes the separation employs the so called "Mirror Model" and is based on adaptive activation functions, whose shape is properly modified during learning. Nonlinear functions involved in the processing of complex signals are realized by pairs of spline neurons called "splitting functions", working on the real and the imaginary part of the signal respectively. Theoretical proof of existence and uniqueness of the solution under proper assumptions is also provided. In particular a simple adaptation algorithm is derived and some experimental results that demonstrate the effectiveness of the proposed solution are shown. [ABSTRACT FROM AUTHOR]

For polarized signals, which arise in many application fields, a statistical framework in terms of quaternionic random processes is proposed. Based on it, the ability of real-, complex- and quaternionic-valued multi-layer perceptrons (MLPs) of performing classification tasks for such signals is evaluated. For the multi-dimensional neural networks the relevance of class label representations is discussed. For signal to noise separation it is shown that the quaternionic MLP yields an optimal solution. Results on the classification of two different polarized signals are also reported. [ABSTRACT FROM AUTHOR]