Statistical mechanics of neural networks
We investigate five different problems in the field of the statistical mechanics of neural networks. The first three problems involve attractor neural networks that optimise particular cost functions for storage of static memories as attractors of the neural dynamics. We study the effects of replica symmetry breaking (RSB) and attempt to find algorithms that will produce the optimal network if error-free storage is impossible. For the Gardner-Derrida network we show that full RSB is necessary for an exact solution everywhere above saturation. We also show that, no matter what the cost function that is optimised, if the distribution of stabilities has a gap then the Parisi replica ansatz that has been made is unstable. For the noise-optimal network we find a continuous transition to replica symmetry breaking at the AT line, in line with previous studies of RSB for different networks. The change to RSBl improves the agreement between "experimental" and theoretical calculations of the local stability distribution ρ(λ) significantly. The effect on observables is smaller. We show that if the network is presented with a training set which has been generated from a set of prototypes by some noisy rule, but neither the noise level nor the prototypes are known, then the perceptron algorithm is the best initial choice to produce a network that will generalise well. If additional information is available more sophisticated algorithms will be faster and give a smaller generalisation error. The remaining problems deal with attractor neural networks with separable interaction matrices which can be used (under parallel dynamics) to store sequences of patterns without the need for time delays. We look at the effects of correlations on a singlesequence network, and numerically investigate the storage capacity of a network storing an extensive number of patterns in such sequences. When correlations are implemented along with a term in the interaction matrix designed to suppress some of the effects of those correlations, the competition between the two produces a rich range of behaviour. Contrary to expectations, increasing the correlations and the operating temperature proves capable of improving the sequenceprocessing behaviour of the network. Finally, we demonstrate that a network storing a large number of sequences of patterns using a Hebb-like rule can store approximately twice as many patterns as the network trained with the Hebb rule to store individual patterns.