Optimal storage capacity of neural networks at finite temperatures

Gardner's analysis of the optimal storage capacity of neural networks is extended to study finite-temperature effects. The typical volume of the space of interactions is calculated for strongly-diluted networks as a function of the storage ratio α, temperature T , and the tolerance parameter m, from which the optimal storage capacity α c is obtained as a function of T and m. At zero temperature it is found that α c = 2 regardless of m while α c in general increases with the tolerance at finite temperatures. We show how the best performance for given α and T is obtained, which reveals a first-order transition from high-quality performance to low-quality one at low temperatures. An approximate criterion for recalling, which is valid near m = 1, is also discussed.