# Angular Regularization for Positive-Unlabeled Learning on the Hypersphere

arXiv: https://arxiv.org/abs/2512.06785 · HTML: https://mlsanity.com/paper/2512.06785/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 43 references that are in the map.
- First posted on arXiv 7 Dec 2025.

## Builds on (most cited first, 21 of 21)

- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Momentum Contrast for Unsupervised Visual Representation Learning](https://mlsanity.com/paper/1911.05722/) (2019)
- [Unsupervised Feature Learning via Non-Parametric Instance-level Discrimination](https://mlsanity.com/paper/1805.01978/) (2018)
- [Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere](https://mlsanity.com/paper/2005.10242/) (2020)
- [Pseudo-Labeling and Confirmation Bias in Deep Semi-Supervised Learning](https://mlsanity.com/paper/1908.02983/) (2019)
- [Positive-Unlabeled Learning with Non-Negative Risk Estimator](https://mlsanity.com/paper/1703.00593/) (2017)
- [Learning from positive and unlabeled data: a survey](https://mlsanity.com/paper/1811.04820/) (2018)
- [Mixture Proportion Estimation via Kernel Embedding of Distributions](https://mlsanity.com/paper/1603.02501/) (2016)
- [Learning with Confident Examples: Rank Pruning for Robust Classification with Noisy Labels](https://mlsanity.com/paper/1705.01936/) (2017)
- [Self-PU: Self Boosted and Calibrated Positive-Unlabeled Training](https://mlsanity.com/paper/2006.11280/) (2020)
- [Class-prior Estimation for Learning from Positive and Unlabeled Data](https://mlsanity.com/paper/1611.01586/) (2016)
- [Dist-PU: Positive-Unlabeled Learning from a Label Distribution Perspective](https://mlsanity.com/paper/2212.02801/) (2022)
- [Classification from Positive, Unlabeled and Biased Negative Data](https://mlsanity.com/paper/1810.00846/) (2018)
- [Learning from Positive and Unlabeled Data with Arbitrary Positive Shift](https://mlsanity.com/paper/2002.10261/) (2020)
- [Directional Statistics in Machine Learning: a Brief Review](https://mlsanity.com/paper/1605.00316/) (2016)
- [Positive Unlabeled Contrastive Learning](https://mlsanity.com/paper/2206.01206/) (2022)
- [Leveraging Ensemble Diversity for Robust Self-Training in the Presence of Sample Selection Bias](https://mlsanity.com/paper/2310.14814/) (2023)
- [Dens-PU: PU Learning with Density-Based Positive Labeled Augmentation](https://mlsanity.com/paper/2303.11848/) (2023)
- [Defect detection using weakly supervised learning](https://mlsanity.com/paper/2303.15092/) (2023)
- [Deep learning based black spot identification on Greek road networks](https://mlsanity.com/paper/2306.10734/) (2023)

## Abstract (arXiv)

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.