  Variational autoencoders employ an encoding neural network to generate a
probabilistic representation of a data set within a low-dimensional space of
latent variables followed by a decoding stage that maps the latent variables
back to the original variable space. Once trained, a statistical ensemble of
simulated data realizations can be obtained by randomly assigning values to the
latent variables that are subsequently processed by the decoding section of the
network. To determine the accuracy of such a procedure when applied to lattice
models, an autoencoder is here trained on a thermal equilibrium distribution of
Ising spin realizations. When the output of the decoder for synthetic data is
interpreted probabilistically, spin realizations can be generated by randomly
assigning spin values according to the computed likelihood. The resulting state
distribution in energy-magnetization space then qualitatively resembles that of
the training samples. However, because correlations between spins are
suppressed, the computed energies are unphysically large for low-dimensional
latent variable spaces. The features of the learned distributions as a function
of temperature, however, provide a qualitative indication of the presence of a
phase transition and the distribution of realizations with characteristic
cluster sizes.
