Why is topology hard to learn?

Phase classification has become a prototypical benchmark for data-driven analysis of condensed matter physics. The type and complexity of the phase transition dictate the level of complexity of the algorithm one has to employ. This topic has been broadly explored, offering a menu of both supervised and unsupervised techniques ranging from simple clustering [1-3] to more complex machine learning methods [4-7]. The phase classification problem is most commonly posed like so: we allow our model to view a dataset that is both relevant and straightforwardly obtainable in the scenario we wish to study. We introduce this data set to a model that has no prior knowledge of underlying physics.