Optimising synaptic learning rules in linear associative memories.

Associative matrix memories with real-valued synapses have been studied in many incarnations. We consider how the signal/noise ratio for associations depends on the form of the learning rule, and we show that a covariance rule is optimal. Two other rules, which have been suggested in the neurobiology literature, are asymptotically optimal in the limit of sparse coding. The results appear to contradict a line of reasoning particularly prevalent in the physics community. It turns out that the apparent conflict is due to the adoption of different underlying models. Ironically, they perform identically at their co-incident optima. We give details of the mathematical results, and discuss some other possible derivations and definitions of the signal/noise ratio.