About the de Almeida–Thouless line in neural networks

L. Albanese, A. Alessandrelli, A. Annibale and A. Barra

Physica A: Statistical Mechanics and its Applications, 2024, vol. 633, issue C

Abstract: 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 de Almeida and Thouless for spin glasses. The method is based on an expansion of the free-energy obtained within one-step of replica symmetry breaking (RSB) around the 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 appendices we test the method on the Sherrington–Kirkpatrick and the Ising 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. Our method provides a generalization of the AT approach, which does not rely on this limit and can be applied to systems with discontinuous phase transitions, as we show explicitly for the spherical P-spin model, recovering the known RS instability line.

Keywords: Neural networks; AT line; de Almeida–Thouless line; RSB; Hopfield model; Dense associative memories (search for similar items in EconPapers)
Date: 2024
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:633:y:2024:i:c:s0378437123009275

DOI: 10.1016/j.physa.2023.129372

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