We examine unsupervised machine learning techniques to learn features that best describe configurations of the two-dimensional Ising model and the three-dimensional $XY$ model. The methods range from principal component analysis over manifold and clustering methods to artificial neural-network-based variational autoencoders. They are applied to Monte Carlo–sampled configurations and have, a priori, no knowledge about the Hamiltonian or the order parameter. We find that the most promising algorithms are principal component analysis and variational autoencoders. Their predicted latent parameters correspond to the known order parameters. The latent representations of the models in question are clustered, which makes it possible to identify phases without prior knowledge of their existence. Furthermore, we find that the reconstruction loss function can be used as a universal identifier for phase transitions.

• Received 13 March 2017
• Revised 9 June 2017

DOI:https://doi.org/10.1103/PhysRevE.96.022140