Linking discrete orthogonality with dilation and translation for incomplete sigma-pi neural networks of Hopfield-type

In this paper, we show how to extend well-known discrete orthogonality results for complete sigma-pi neural networks on bipolar coded information in presence of dilation and translation of the signals. The approach leads to a whole family of functions being able to implement any given Boolean function. Unfortunately, the complexity of such complete higher order neural network realizations increases exponentially with the dimension of the signal space. Therefore, in practise one often only considers incomplete situations accepting that not all but hopefully the most relevant information or Boolean functions can be realized. At this point, the introduced dilation and translation parameters play an essential rôle because they can be tuned appropriately in order to fit the concrete representation problem as best as possible without any significant increase of complexity. In detail, we explain our approach in context of Hopfield-type neural networks including the presentation of a new learning algorithm for such generalized networks.