ENERGY LANDSCAPE OF NEURAL NETWORKS STORING SPATIALLY CORRELATED PATTERNS

Hopfield-like neural networks with spatially organized data are studied by a mean-field theory. The internal structure of the data is described by a matrix (C) over cap whose elements C-ij are equal to the correlation between two pixels, i and j, of any input pattern. The model considered here is described by the matrix (C) over cap in which the pixel-pixel correlation is the same for all pairs of pixels and is equal to lambda/N. The statistical properties of the model depend on three parameters: the reduced number of the stored patterns alpha, the temperature T and the reduced number of strength correlations of the pixels lambda. The phase diagram in the space of parameters lambda and alpha at temperature T = 0 is obtained. The network can retrieve patterns at T = 0 for alpha < alpha(c), where alpha(c) similar or equal to 0.14 as for the usual Hopfield neural network, but there is a new transition line above which a new local minimum of the free energy arises. This minimum corresponds to a ferromagnetic ordering of the neurons. There is another additional minimum (between the next two lines) that corresponds to mixed ordering. We also find the region where the ferromagnetic state becomes the ground state of the system