THE CAPACITY OF THE HOPFIELD ASSOCIATIVE MEMORY

Techniques for coding theory are applied to study the capacity of the Hopfield associative memory. Such a memory stores n-tuple of plus or minus 1's. The components change depending on a hardlimited version of linear functions of all other components. With symmetric connections between components, a stable state is reached. By building up the connection matrix as a sum-of-outer products of m fundamental memories, a certain one of the m memories is recovered by using an initial n-tuple probe vector less than a Hamming distance n/2 away from the fundamental memory. If m memories are chosen at random, the maximum asymptotic value of m in order that most of the m original memories are recoverable is n/(2 log n). With the added restriction that every one of the m fundamental memories be recoverable, m can be no more than n(4 log n) asymptotically as n approaches infinity. Extensions are considered, in particular to capacity under quantization of the outer-product connection matrix. This quantized memory capacity problem is related to the capacity of the quantized Gaussian channel.