Neural System Learning on Complex-Valued Manifolds

An instance of artificial neural learning is by criterion optimization, where the criterion to optimize measures the learning ability of the neural network either in supervised learning (the adaptation is supervised by a teacher) or in unsupervised learning (the adaptation of network parameters proceeds on the basis of the information that the neural system is able to extract from the inputs). In some circumstances of interest, the space of parameters of the neural system is restricted to a particular feasible space via suitable bounds, which represent the constraint imposed by the learning problem at hand. In this case, the optimization rules to adapt the parameters of the neural network must be designed according to the known constraints. If the set of feasible parameters form a smooth continuous set, namely, a differentiable manifold, the design of adaptation rules falls in the realm of differential geometrical methods for neural networks and learning and of the numerical geometric integration of learning equations. The present chapter deals with complex-valued parameter-manifolds and with applications of complex-valued artificial neural networks whose connection-parameters live in complex-valued manifolds. The successful applications of such neural networks, which are described within the present chapter, are to blind source separation of complex-valued sources and to multichannel blind deconvolution of signals in telecommunications, to nondestructive evaluation of materials in industrial metallic slabs production and to the purely algorithmic problem of averaging the parameters of a pool of cooperative complex-valued neural networks. The present chapter recalls those notions of differential geometry that are instrumental in the definition of a consistent learning theory over complex-valued differentiable manifolds and introduces some learning problems and their solutions.