Eigenvector dreaming

Among the performance-enhancing procedures for Hopfield-type networks that implement associative memory, Hebbian unlearning (HU) (or dreaming) strikes for its simplicity and lucid biological interpretation. However, it does not easily lend to a clear analytical understanding. Here, we show how HU can be efficiently described in terms of the evolution of the spectrum and the eigenvectors (EVs) of the coupling matrix. That is, we find that HU barely changes the EVs of the coupling matrix, whereas the benefits of the procedure can be ascribed to an intuitive evolution of the spectrum. We use these ideas to design novel dreaming algorithms that are effective from a computational point of view and are analytically far more transparent than the original scheme.