On the Computational Complexity of Learning Bithreshold Neural Units and Networks

Abstract

We study the questions concerning the properties and capabilities of computational bithreshold real-weighted neural-like units. We give and justify the two sufficient conditions ensuring the possibility of separation of two sets in n-dimensional vector space by means of one bithreshold neuron. Our approach is based on application of convex and affine hulls of sets and is feasible in the case when one of the two sets is a compact and the second one is finite. We also correct and refine some previous results concerning bithreshold separability. Then the hardness of the learning bithreshold neurons is considered. We examine the complexity of the problem of checking whether the given Boolean function of n variables can be realizable by single bithreshold unit. Our main result is that the problem of verifying the bithreshold separability is NP-complete. The same is true for neural networks consisting of such computational units. We propose some continuous modifications of the bithreshold activation function to smooth away these difficulties and to make possible the application of modern paradigms and learning techniques for such networks.