Memory and optimisation in neural network models
A numerical study of two classes of neural network models is presented.
The performance of Ising spin neural networks as content-addressable memories
for the storage of bit patterns is analysed. By studying systems of increasing
sizes, behaviour consistent with fintite-size scaling, characteristic of a first-order
phase transition, is shown to be exhibited by the basins of attraction of the
stored patterns in the Hopfield model. A local iterative learning algorithm is
then developed for these models which is shown to achieve perfect storage of
nominated patterns with near-optimal content-addressability. Similar scaling
behaviour of the associated basins of attraction is observed. For both this learning
algorithm and the Hopfield model, by extrapolating to the thermodynamic
limit, estimates are obtained for the critical minimum overlap which an input
pattern must have with a stored pattern in order to successfully retrieve it.
The role of a neural network as a tool for optimising cost functions of binary valued
variables is also studied. The particular application considered is that of
restoring binary images which have become corrupted by noise. Image restorations
are achieved by representing the array of pixel intensities as a network
of analogue neurons. The performance of the network is shown to compare
favourably with two other deterministic methods-a gradient descent on the
same cost function and a majority-rule scheme-both in terms of restoring images
and in terms of minimising the cost function.
All of the computationally intensive simulations exploit the inherent parallelism
in the models: both SIMD (the ICL DAP) and MIMD (the Meiko Computing
Surface) machines are used.