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Alien attractors and memory annihilation of structured sets in Hopfield networks

This paper considers the encoding of structured sets into Hopfield associative memories. A structured set is a set of vectors with equal Hamming distance h from one another, and its centroid is an external vector 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 stable 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.