Angular Regularization for Positive-Unlabeled Learning on the Hypersphere

Abstract: Positive-Unlabeled (PU) learning addresses classification problems where only a subset of positive examples is labeled and the remaining data is unlabeled, making explicit negative supervision unavailable. Existing PU methods often rely on negative-risk estimation or pseudo-labeling, which either require strong distributional assumptions or can collapse in high-dimensional settings. We propose AngularPU, a novel PU framework that operates on the unit hypersphere using cosine similarity and angular margin. In our formulation, the positive class is represented by a learnable prototype vector, and classification reduces to thresholding the cosine similarity between an embedding and this prototype-eliminating the need for explicit negative modeling. To counteract the tendency of unlabeled embeddings to cluster near the positive prototype, we introduce an angular regularizer that encourages dispersion of the unlabeled set over the hypersphere, improving separation. We provide theoretical guarantees on the Bayes-optimality of the angular decision rule, consistency of the learned prototype, and the effect of the regularizer on the unlabeled distribution. Experiments on benchmark datasets demonstrate that AngularPU achieves competitive or superior performance compared to state-of-the-art PU methods, particularly in settings with scarce positives and high-dimensional embeddings, while offering geometric interpretability and scalability.