Variational Autoencoder Analysis of Ising Model Statistical Distributions and Phase Transitions

Variational autoencoders employ a neural network to encode a probabilistic representation of a data set onto a low-dimensional space of latent variables. A second decoding stage then maps the latent variables back to the original variable space. Once trained, a statistical ensemble of simulated data realizations can be obtained by decoding random sets of latent variables. To determine the accuracy of this procedure in the context of lattice models, an autoencoder is trained on a thermal equilibrium distribution of Ising spin realizations. Synthetic spin realizations are then obtained by decoding sets of randomly assigned latent variable values and interpreting the output as the likelihood of a certain spin orientation. The resulting state distribution in energy-magnetization space then qualitatively resembles that of the training samples. However, this paper demonstrates that because such techniques suppress correlations among spins, the computed energies are unphysically large for low-dimensional latent variable spaces. The features of the learned distributions as a function of temperature, however, qualitatively indicate the presence of phase transitions.