An energy function-based design method for discrete hopfield associative memory with attractive fixed points

An energy function-based autoassociative memory design method to store a given set of unipolar binary memory vectors as attractive fixed points of an asynchronous discrete Hopfield network (DHN) is presented. The discrete quadratic energy function whose local minima correspond to the attractive fixed points of the network is constructed via solving a system of linear inequalities derived from the strict local minimality conditions. The weights and the thresholds are then calculated using this energy function. If the inequality system is infeasible, we conclude that no such asynchronous DHN exists, and extend the method to design a discrete piecewise quadratic energy function, which can be minimized by a generalized version of the conventional DHN, also proposed herein. In spite of its computational complexity, computer simulations indicate that the original method performs better than the conventional design methods in the sense that the memory can store, and provide the attractiveness for almost all memory sets whose cardinality is less than or equal to the dimension of its elements. The overall method, together with its extension, guarantees the storage of an arbitrary collection of memory vectors, which are mutually at least two Hamming distances away from each other, in the resulting network.