Robuste Lernverfahren bei schlecht konditionierten Problemstellungen für neuronale PCA- und ICA-Netzwerke mit Adaption der Lernraten

Robust learning rules in case of ill-conditioned problems for neural PCAand ICA-networks with adaptation of the learning rates Methods for Blind Signal Processing (BSP) are becoming noticeably more important. Amongst others, these techniques are applied in biomedical signal processing, image processing and speech processing. Within the BSP, the Principal Component Analysis (PCA) and the Independent Component Analysis (ICA) are of particular importance. Considering the vector x(t) of the stochastic input processes, a principal component is referred to as the inner product, which consists of x(t) and an eigenvector of the autocorrelation matrix Rxx of x(t). On the other hand, the linear ICA is based on the assumption that x(t) is the result of a linear transformation. This transformation is described by the vector s(t) of the independent stochastic source processes and the socalled mixing-matrix A. Neither the stochastic source processes, nor the mixing-matrix A are known. In this regard, adaptive systems formed by neural networks enable a massively parallel extraction of principal components and source processes, respectively. In order to include continuous-time implementations in addition to discrete-time implementations, the neural networks are described by continuous-time systems below. An attractive feature of neural systems is their ability to adapt themselves to special environmental conditions by changing their weights. This procedure of adaptation of parameters is called learning and is carried out by a unit which processes the input vector in continuoustime case. This unit which is described by a system of differential equations represents the implementation of the so-called learning rule. Due to parallel operation of neural networks, the necessary time of learning is primarily determined by conditioning of the optimization problem. Concerning PCA and ICA, the conditioning is given by the condition number of Rxx and A, respectively, as well as the stochastic properties of the principal components and the source processes, respectively. Accordingly, the main objective will be the generation of learning rules for PCA and ICA networks being robust with respect to these characteristics. By means of the so-called cross-entropy, the conditions related to stochastic properties of output processes of PCA and ICA are transferred into an estimation task of the density function of the vectorial stochastic process x(t). Through this approach the