Machine learning the Ising transition: A comparison between discriminative and generative approaches

The detection of phase transitions is a central task in many-body physics. To automate this process, the task can be phrased as a classification problem. Classification problems can be approached in two fundamentally distinct ways: through either a discriminative or a generative method. In general, it is unclear which of these two approaches is most suitable for a given problem. In this work, we answer the question of how one should approach the detection of the thermal transition within the classical two-dimensional square-lattice ferromagnetic Ising model by performing a systematic numerical case study. We find that in a data-driven setting with limited system knowledge, neural network-based approaches outperform nonparametric generative methods in terms of the achieved accuracy for a given dataset size. Both parametric generative and discriminative approaches may be viable depending on how the classification task is set up. Conversely, in a knowledge-driven setting where the system Hamiltonian is known, nonparametric generative approaches leveraging sufficient statistics are superior in terms of the accuracy reached at a given budget of computation time for all but the smallest sample sizes or highest error tolerances. Our findings provide a guide for the selection of machine-learning strategies for phase transition detection based on criteria such as the amount of available system knowledge, the number of data points, or computational constraints.