Highly Symmetric Neural Networks of Hopfield Type (exact results)

A set of fixed points of the Hopfield type neural network is under investigation. Its connection matrix is constructed with regard to the Hebb rule from a highly symmetric set of the memorized patterns. Depending on the external parameter the analytic description of the fixed points set has been obtained. A set of fixed points of the Hopfield type neural network is under investigation. Its connection matrix is constructed with regard to the Hebb rule from a (p × n)-matrix S of memorized patterns: Here n is the number of neurons, p is the number of memorized patterns s (l) , which are the rows of the matrix S, and x is an arbitrary real number. Depending on x the memorized patterns s (l) are interpreted as p distorted vectors of the standard ε(n) = (1, 1,. .. , 1 n). (1) The problem is as follows: the network has to be learned by p-times showing of the standard (1), but a distortion has slipped in the learning process. How does the fixed points set depends on the value of this distortion x? Depending on the distortion parameter x the analytic description of the fixed points set has been obtained. It turns out to be very important that the memorized patterns s (l) form a highly symmetric group of vectors: all of them correlate one with another in the same way: (s (l) , s (l ′)) = r(x), (2) where r(x) is independent of l, l ′ = 1, 2,. .. , p. Namely this was the reason to use the words " highly symmetric " in the title. It is known [1], that the fixed points of a network of our kind have to be of the form: