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

Let (w/sub ij/) be the weights of the connections of a neural network with n nodes, calculated from m data vectors v/sup 1/, ..., v/sup m/ in (1,-1)/sup n/, according to the Hebb rule. The author proves that if m is not too large relative to n and the v/sup k/ are random, then the w/sub ij/ constitute, with high probability, a perfect representation of the v/sup k/ in the sense that the v/sup k/ are completely determined by the w/sub 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/sup k/ can actually be recovered by letting the network evolve until equilibrium is attained. In the specific case where the entries of the v/sup 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.<
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