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Distributions of data or sensory stimuli often enjoy underlying invariances.How and to what extent those symmetries are captured by unsupervised learningmethods is a relevant question in machine learning and in computationalneuroscience. We study here, through a combination of numerical and analyticaltools, the learning dynamics of Restricted Boltzmann Machines (RBM), a neuralnetwork paradigm for representation learning. As learning proceeds from arandom configuration of the network weights, we show the existence of, andcharacterize a symmetry-breaking phenomenon, in which the latent variablesacquire receptive fields focusing on limited parts of the invariant manifoldsupporting the data. The symmetry is restored at large learning times throughthe diffusion of the receptive field over the invariant manifold; hence, theRBM effectively spans a continuous attractor in the space of network weights.This symmetry-breaking phenomenon takes place only if the amount of dataavailable for training exceeds some critical value, depending on the networksize and the intensity of symmetry-induced correlations in the data; below this'retarded-learning' threshold, the network weights are essentially noisy andoverfit the data.

journal of physics. a, mathematical and theoretical

'Place-cell' emergence and learning of invariant data with restricted  Boltzmann machines: breaking and dynamical restoration of continuous  symmetries in the weight space