We discuss two solvable grokking (generalisation beyond overfitting) models in a rule-learning scenario. We show that grokking is a phase transition and find exact analytic expressions for the critical exponents, grokking probability, and grokking time distribution. Further, we introduce a tensor network map that connects the proposed grokking setup with the standard (perceptron) statistical learning theory and provide evidence that grokking is a consequence of the locality of the teacher model. We analyze the rule-30 cellular automaton learning task, numerically determine the critical exponent and the grokking time distribution, and compare them with the prediction of the proposed grokking model. Finally, we numerically study the connection between structure formation and grokking.

Positive unlabeled learning is a binary classification problem with positive and unlabeled data. It is common in domains where negative labels are costly or impossible to obtain, e.g., medicine and personalized advertising. Most approaches to positive unlabeled learning apply to specific data types (e.g., images, categorical data) and can not generate new positive and negative samples. This work introduces a feature-space distance-based tensor network approach to the positive unlabeled learning problem. The presented method is not domain specific and significantly improves the state-of-the-art results on the MNIST image and 15 categorical/mixed datasets. The trained tensor network model is also a generative model and enables the generation of new positive and negative instances.