The replica-symmetric solution without replica trick for the Hopfield model

DOI10.1007/BF02188221zbMath0835.60092MaRDI QIDQ1897016

Brunello Tirozzi, Mariya Shcherbina, Leonid Pastur

Publication date: 8 April 1996

Published in: Journal of Statistical Physics (Search for Journal in Brave)

zbMATH Keywords

perturbative expansionHopfield modelself-averaging propertyreplica-symmetry saddle-point equations

Mathematics Subject Classification ID

Other physical applications of random processes (60K40) Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics (82C44)

Related Items (24)

Gibbs states of the Hopfield model in the regime of perfect memory ⋮ On the replica symmetric equations for the Hopfield model ⋮ Strong replica symmetry in high-dimensional optimal Bayesian inference ⋮ Hierarchical neural networks perform both serial and parallel processing ⋮ An analysis on neural dynamics with saturated sigmoidal functions ⋮ A new mechanical approach to handle generalized Hopfield neural networks ⋮ Dreaming neural networks: rigorous results ⋮ Generalized Guerra's interpolation schemes for dense associative neural networks ⋮ Non-convex multi-species Hopfield models ⋮ Fluctuations of the free energy in the high temperature Hopfield model ⋮ The replica symmetric approximation of the analogical neural network ⋮ Spontaneous Replica Symmetry Breaking and Interpolation Methods for Complex Statistical Mechanics Systems ⋮ About the ergodic regime in the analogical Hopfield neural networks: Moments of the partition function ⋮ Large deviation principles for the Hopfield model and the Kac-Hopfield model ⋮ The relativistic Hopfield model with correlated patterns ⋮ The adaptive interpolation method: a simple scheme to prove replica formulas in Bayesian inference ⋮ Unnamed Item ⋮ Unnamed Item ⋮ Free energies of Boltzmann machines: self-averaging, annealed and replica symmetric approximations in the thermodynamic limit ⋮ Gibbs states of the Hopfield model with extensively many patterns. ⋮ Free energy and some sample path properties of a random walk with random potential. ⋮ Techniques of replica symmetry breaking and the storage problem of the McCulloch-Pitts neuron ⋮ On the volume of the intersection of a sphere with random half spaces ⋮ Replica symmetry breaking in neural networks: a few steps toward rigorous results

Cites Work

• The free energy of a class of Hopfield models
• Neural networks and physical systems with emergent collective computational abilities.
• Neurons with graded response have collective computational properties like those of two-state neurons.

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