The capacity of associative memory by using r-order shifting rules

The second-order Hopfield associative memory has storage capacity of order O(n/log n), if the stored vectors and probe vector are subject to uniform distributions. Unfortunately, this is not always the case practically. We prove that the capacity drops to order of zero when stored vectors and probe vector have nonuniform distributions. Therefore, it is necessary to use the "shifting" method in the modified Hopfield model to improve the storage capacity, where "shifting" means subtracting means from vectors. The influence of probability distribution on capacity is expressed