Convergent unlearning algorithm for the Hopfield neural network

We investigate asymptotic behaviour of synaptic matrix iterated according to the unlearning algorithm (A. Yu et al., 1994). The algorithm has been proven to converge to the projector (pseudo inverse) rule matrix if the unlearning strength parameter ε>0 does not exceed some critical value. Asymptotic behaviour of normalized synaptic matrix J˜ is considered, relating it to the corresponding spectrum dynamics. It is found that the algorithm converges for arbitrary value of ε, and there are only three possibilities for limiting behaviour of J˜. The first one is successful unlearning which implies the convergence to the projection matrix onto the linear subspace L spanned by maximal subset of linearly independent patterns. At sufficiently large values of ε the typical result of iterations will be failed unlearning, with J˜ converging to the minus projector on random unity vector ξ∈L. We show that failed unlearning results in total memory breakdown. There is also an “intermediate” case when J˜ converges to the projection matrix on some subspace of L. Probability for different asymptotics to appear depending upon unlearning strength is studied for the case of unbiased random patterns. Retrieval properties of the system equipped with limiting synaptic matrix are also discussed