On the number of memories that can be perfectly stored in a neural net with Hebb weights

Let {w_ij} be the weights of the connections of a neural network with n nodes, calculated from m data vectors v¹, ···, v^m in {1,-1}ⁿ, according to the Hebb rule. The author proves that if m is not too large relative to n and the v^k are random, then the w_ij constitute, with high probability, a perfect representation of the v^k in the sense that the v ^k are completely determined by the w_ij up to their sign. The conditions under which this is established turn out to be less restrictive than those under which it has been shown that the v^k can actually be recovered by letting the network evolve until equilibrium is attained. In the specific case where the entries of the v^k are independent and equal to 1 or -1 with probability 1/2, the condition on m is that m should not exceed n/0.7 log n