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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Statistical mechanics of a multiconnected Hopfield neural-network model in a transverse field

Yu‐qiang Ma, Chang-De Gong

Published Feb 1, 1995• 5 citations
Publisher

Abstract

The Hopfield neural-network model with p-spin interactions in the presence of a transverse field is introduced and solved exactly in the limit p\ensuremath{\rightarrow}\ensuremath{\infty}. In the phase diagrams drawn as a function of the temperature, the important results such as reentrance are found, and the effects of the quantum fluctuations on the phase transitions, the retrieval phase, and the storage ratio \ensuremath{\alpha} are examined.

DOI
10.1103/physreve.51.1573

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The Hopfield neural-network model with p-spin interactions in the presence of a transverse field is introduced and solved exactly in the limit p\ensuremath{\rightarrow}\ensuremath{\infty}. In the phase diagrams drawn as a function of the temperature, the important results such as reentrance are found, and the effects of the quantum fluctuations on the phase transitions, the retrieval phase, and the storage ratio \ensuremath{\alpha} are examined.
We examine a quantum Hopfield neural-network model in the presence of trimodal random transverse fields and random neuronal thresholds within the method of statistical physics. We use the Trotter decomposition to map the problem into an equivalent classical random Hopfield-type Ising model and obtain phase transitions between the ferromagnetic retrieval and the paramagnetic phases. The influence of competition between the diluted random transverse fields and the diluted random thresholds on the system is discussed, and some interesting results such as tricritical points and reentrance are analyzed.
The Hopfield model in a transverse field is investigated in order to clarify how quantum fluctuations affect the macroscopic behavior of neural networks. Using the Trotter decomposition and the replica method, we find that the ααα (the ratio of the number of stored patterns to the system size)-ΔΔΔ (the strength of the transverse field) phase diagram of this model in the ground state resembles the ααα-TTT phase diagram of the Hopfield model quantitatively, within the replica-symmetric and static approximations. This fact suggests that quantum fluctuations play quite similar roles to thermal fluctuations in neural networks as long as macroscopic properties are concerned.
We develop the statistical mechanics of the Hopfield model in a transverse field to investigate how quantum fluctuations affect the macroscopic behavior of neural networks. When the number of embedded patterns is finite, the Trotter decomposition reduces the problem to that of a random Ising model. It turns out that the effects of quantum fluctuations on macroscopic variables play the same roles as those of thermal fluctuations. For an extensive number of embedded patterns, we apply the replica method to the Trotter-decomposed system. The result is summarized as a ground-state phase diagram drawn in terms of the number of patterns per site, ααα, and the strength of the transverse field, ΔΔΔ. The phase diagram coincides very accurately with that of the conventional classical Hopfield model if we replace the temperature T in the latter model by ΔΔΔ. Quantum fluctuations are thus concluded to be quite similar to thermal fluctuations in determination of the macroscopic behavior of the present model.
A quantum Hopfield spin-glass model with transverse fields is presented. A set of equations for the order parameters of a large class of spin-glass models with transverse fields is obtained, and the phase transitions are analyzed with a finite number of patterns p. We find that a reentrant phenomenon is possible for a quantum Mattis model with transverse fields, contrary to the quantum transverse Sherrington-Kirkpatrick-like model. The effects of quantum fluctuations (induced by the transverse field) on the ordered phases of systems with competing interactions are also discussed.