Journal of Statistical Physics

, Volume 172, Issue 5, pp 1247–1269 | Cite as

Non-convex Multi-species Hopfield Models

  • Elena AgliariEmail author
  • Danila Migliozzi
  • Daniele Tantari


In this work we introduce a multi-species generalization of the Hopfield model for associative memory, where neurons are divided into groups and both inter-groups and intra-groups pair-wise interactions are considered, with different intensities. Thus, this system contains two of the main ingredients of modern deep neural-network architectures: Hebbian interactions to store patterns of information and multiple layers coding different levels of correlations. The model is completely solvable in the low-load regime with a suitable generalization of the Hamilton–Jacobi technique, despite the Hamiltonian can be a non-definite quadratic form of the Mattis magnetizations. The family of multi-species Hopfield model includes, as special cases, the 3-layers Restricted Boltzmann Machine with Gaussian hidden layer and the Bidirectional Associative Memory model.


Neural networks Boltzmann machines Multipartite models 



E.A. acknowledges financial support by Sapienza Università di Roma (Project No. RG11715C7CC31E3D). D.T. is supported by Scuola Normale Superiore and National Group of Mathematical Physics GNFM-INdAM.


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Authors and Affiliations

  1. 1.Dipartimento di MatematicaSapienza Università di RomaRomeItaly
  2. 2.Scuola Normale SuperiorePisaItaly

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