Gauged neural network: Phase structure, learning, and associative memory

Abstract

A gauge model of neural network is introduced, which resembles the Z(2) Higgs lattice gauge theory of high-energy physics. It contains a neuron variable S x = ± 1 on each site x of a 3D lattice and a synaptic-connection variable J x μ = ± 1 on each link ( x , x + μ ^ ) ( μ = 1 , 2 , 3 ) . The model is regarded as a generalization of the Hopfield model of associative memory to a model of learning by converting the synaptic weight between x and x + μ ^ to a dynamical Z(2) gauge variable J x μ . The local Z(2) gauge symmetry is inherited from the Hopfield model and assures us the locality of time evolutions of S x and J x μ and a generalized Hebbian learning rule. At finite “temperatures”, numerical simulations show that the model exhibits the Higgs, confinement, and Coulomb phases. We simulate dynamical processes of learning a pattern of S x and recalling it, and classify the parameter space according to the performance. At some parameter regions, stable column-layer structures in signal propagations are spontaneously generated. Mutual interactions between S x and J x μ induce partial memory loss as expected.