ALIEN ATTRACTORS AND MEMORY ANNIHILATION OF STRUCTURED SETS IN HOPFIELD NETWORKS

This paper considers the encoding of structured sets into Hopfield ass ociative memories. A structured set is a set of vectors with equal Ham ming distance h from one another, and its centroid is an external vect or that has distance h/2 from every vector of the set. Structured sets having centroids are not infrequent. When such a set is encoded into a noiseless Hopfield associative memory using a bipolar outer-product connection matrix, and the network operates with synchronous neuronal update, the memory of all encoded vectors is annihilated even for sets with as few as three vectors in dimension n > 5 (four for n = 5). In such self-annihilating structured sets, the centroid emerges as a stab le attractor. We call it an alien attractor. For canonical structured sets, self-annihilation takes place only if h < n/2. Self-annihilation does not occur and alien attractors do not emerge in dimensions less than five.