On the Central Limit Theorem for the Overlap in the Hopfield Model
Barbara Gentz · Birkhäuser Boston eBooks · 1998
We consider the Hopfield model with N neurons and an increasing number M = M ( N ) of randomly chosen patterns. Under the condition M 2 / N → 0, we prove for every fixed choice of overlap parameters a central limit theorem as N → ∞, which holds for almost all realizations of the random patterns. In the special case where the temperature is above the critical one and there is no external magnetic field, the condition M 3/2 log M ≤ N suffices. As in the case of a finite number of patterns, the central limit theorem requires a centering which depends on the random patterns. In addition, we describe the almost sure asymptotic behavior of the partition function under the condition M 3 / N → 0. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.