About the de Almeida-Thouless line in neural networks

Linda Albanese, Adriano Barra, Alessia Annibale

Published 2023-03-11Version 1

In this work we present a rigorous and straightforward method to detect the onset of the instability of replica-symmetric theories in information processing systems, which does not require a full replica analysis as in the method originally proposed by Almeida-Thouless for spin glasses. The method is based on an expansion of the free-energy obtained within one-step of the replica symmetry breaking (RSB) scheme around the replica-symmetric (RS) value. As such, it requires solely continuity and differentiability of the free-energy and it is robust to be applied broadly to systems with quenched disorder. We apply the method to the Hopfield model and to neural networks with multi-node Hebbian interactions, as case studies. In the appendix we test the method on the Sherrington-Kirkpatrick, the Ising P-spin and the spherical P-spin models, recovering the AT lines known in the literature for these models, as a special limit, which corresponds to assuming that the transition from the RS to the RSB phase can be obtained by varying continuously the order parameters. Therefore, our method provides a generalization of the AT approach, which can be applied to systems with a discontinuous transition.