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
\pm 1
's. 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, one hopes to be able 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 asympotic 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.