# Gauge Neural Network with Z(2) Synaptic Variables: Phase Structure and Simulation of Learning and Recalling Patterns

Gauge Neural Network with Z(2) Synaptic Variables: Phase Structure and Simulation of Learning and... We study the Z(2) gauge-invariant neural network which is defined on a partially connected random network and involves Z(2) neuron variables $$S_i$$ S i ( $$=\pm$$ = ± 1) and Z(2) synaptic connection (gauge) variables $$J_{ij}$$ J i j ( $$=\pm$$ = ± 1). Its energy consists of the Hopfield term $$-c_1S_iJ_{ij}S_j$$ - c 1 S i J i j S j , double Hopfield term $$-c_2 S_iJ_{ij}J_{jk} S_k$$ - c 2 S i J i j J j k S k , and the reverberation (triple Hopfield) term $$-c_3 J_{ij}J_{jk}J_{ki}$$ - c 3 J i j J j k J k i of synaptic self interactions. For the case $$c_2=0$$ c 2 = 0 , its phase diagram in the $$c_3-c_1$$ c 3 - c 1 plane has been studied both for the symmetric couplings $$J_{ij}=J_{ji}$$ J i j = J j i and asymmetric couplings ( $$J_{ij}$$ J i j and $$J_{ji}$$ J j i are independent); it consists of the Higgs, Coulomb and confinement phases, each of which is characterized by the ability of learning and/or recalling patterns. In this paper, we consider the phase diagram for the case of nonvanishing $$c_2$$ c 2 , and examine its effect. We find that the $$c_2$$ c 2 term enlarges the region of Higgs phase and generates a new second-order transition. We also simulate the dynamical process of learning patterns of $$S_i$$ S i and recalling them and measure the performance directly by overlaps of $$S_i$$ S i . We discuss the difference in performance for the cases of Z(2) variables and real variables for synaptic connections. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Neural Processing Letters Springer Journals

# Gauge Neural Network with Z(2) Synaptic Variables: Phase Structure and Simulation of Learning and Recalling Patterns

, Volume 47 (3) – Aug 3, 2017
14 pages

/lp/springer_journal/gauge-neural-network-with-z-2-synaptic-variables-phase-structure-and-rlbnmA5MdD
Publisher
Springer US
Subject
Computer Science; Artificial Intelligence (incl. Robotics); Complex Systems; Computational Intelligence
ISSN
1370-4621
eISSN
1573-773X
D.O.I.
10.1007/s11063-017-9678-3
Publisher site
See Article on Publisher Site

### Abstract

We study the Z(2) gauge-invariant neural network which is defined on a partially connected random network and involves Z(2) neuron variables $$S_i$$ S i ( $$=\pm$$ = ± 1) and Z(2) synaptic connection (gauge) variables $$J_{ij}$$ J i j ( $$=\pm$$ = ± 1). Its energy consists of the Hopfield term $$-c_1S_iJ_{ij}S_j$$ - c 1 S i J i j S j , double Hopfield term $$-c_2 S_iJ_{ij}J_{jk} S_k$$ - c 2 S i J i j J j k S k , and the reverberation (triple Hopfield) term $$-c_3 J_{ij}J_{jk}J_{ki}$$ - c 3 J i j J j k J k i of synaptic self interactions. For the case $$c_2=0$$ c 2 = 0 , its phase diagram in the $$c_3-c_1$$ c 3 - c 1 plane has been studied both for the symmetric couplings $$J_{ij}=J_{ji}$$ J i j = J j i and asymmetric couplings ( $$J_{ij}$$ J i j and $$J_{ji}$$ J j i are independent); it consists of the Higgs, Coulomb and confinement phases, each of which is characterized by the ability of learning and/or recalling patterns. In this paper, we consider the phase diagram for the case of nonvanishing $$c_2$$ c 2 , and examine its effect. We find that the $$c_2$$ c 2 term enlarges the region of Higgs phase and generates a new second-order transition. We also simulate the dynamical process of learning patterns of $$S_i$$ S i and recalling them and measure the performance directly by overlaps of $$S_i$$ S i . We discuss the difference in performance for the cases of Z(2) variables and real variables for synaptic connections.

### Journal

Published: Aug 3, 2017

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