The capacity of the Hopfield associative memory

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