# Learning from Positive and Unlabeled Data with Arbitrary Positive Shift

arXiv: https://arxiv.org/abs/2002.10261 · HTML: https://mlsanity.com/paper/2002.10261/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Feb 2020.

## Built on by (most cited first, 23 of 23)

- [Multiscale Positive-Unlabeled Detection of AI-Generated Texts](https://mlsanity.com/paper/2305.18149/) (2023)
- [Dist-PU: Positive-Unlabeled Learning from a Label Distribution Perspective](https://mlsanity.com/paper/2212.02801/) (2022)
- [Multi-label Classification with Partial Annotations using Class-aware Selective Loss](https://mlsanity.com/paper/2110.10955/) (2021)
- [Beyond Myopia: Learning from Positive and Unlabeled Data through Holistic Predictive Trends](https://mlsanity.com/paper/2310.04078/) (2023)
- [A General Framework for Learning from Weak Supervision](https://mlsanity.com/paper/2402.01922/) (2024)
- [Learning from positive and unlabeled examples -Finite size sample bounds](https://mlsanity.com/paper/2507.07354/) (2025)
- [Dens-PU: PU Learning with Density-Based Positive Labeled Augmentation](https://mlsanity.com/paper/2303.11848/) (2023)
- [DialogVCS: Robust Natural Language Understanding in Dialogue System Upgrade](https://mlsanity.com/paper/2305.14751/) (2023)
- [Uncertainty-aware Pseudo-label Selection for Positive-Unlabeled Learning](https://mlsanity.com/paper/2201.13192/) (2022)
- [Positive-Unlabeled Domain Adaptation](https://mlsanity.com/paper/2202.05695/) (2022)
- [InfoNCE is a Free Lunch for Semantically guided Graph Contrastive Learning](https://mlsanity.com/paper/2505.06282/) (2025)
- [Deep Positive-Unlabeled Anomaly Detection for Contaminated Unlabeled Data](https://mlsanity.com/paper/2405.18929/) (2024)
- [Meta-learning for Positive-unlabeled Classification](https://mlsanity.com/paper/2406.03680/) (2024)
- [Positive-Unlabeled Diffusion Models for Preventing Sensitive Data Generation](https://mlsanity.com/paper/2503.03789/) (2025)
- [Prior shift estimation for positive unlabeled data through the lens of kernel embedding](https://mlsanity.com/paper/2502.21194/) (2025)
- [Improving State-of-the-Art in One-Class Classification by Leveraging Unlabeled Data](https://mlsanity.com/paper/2203.07206/) (2022)
- [Robust and Efficient Imbalanced Positive-Unlabeled Learning with Self-supervision](https://mlsanity.com/paper/2209.02459/) (2022)
- [Positive Unlabeled Learning Selected Not At Random (PULSNAR): class proportion estimation when the SCAR assumption does not hold](https://mlsanity.com/paper/2303.08269/) (2023)
- [Regression with Sensor Data Containing Incomplete Observations](https://mlsanity.com/paper/2304.13415/) (2023)
- [Angular Regularization for Positive-Unlabeled Learning on the Hypersphere](https://mlsanity.com/paper/2512.06785/) (2025)
- [Generalized Distribution-Free Semi-Supervised Learning with Risk Rewrite](https://mlsanity.com/paper/2607.11947/) (2026)
- [Importance Weighting for Unlabeled-unlabeled Learning under Distribution Shift](https://mlsanity.com/paper/2609.10994/) (2026)
- [Partial AUC Maximization from Positive-unlabeled Data](https://mlsanity.com/paper/2610.00284/) (2026)

## Abstract (arXiv)

Positive-unlabeled (PU) learning trains a binary classifier using only positive and unlabeled data. A common simplifying assumption is that the positive data is representative of the target positive class. This assumption rarely holds in practice due to temporal drift, domain shift, and/or adversarial manipulation. This paper shows that PU learning is possible even with arbitrarily non-representative positive data given unlabeled data from the source and target distributions. Our key insight is that only the negative class's distribution need be fixed. We integrate this into two statistically consistent methods to address arbitrary positive bias - one approach combines negative-unlabeled learning with unlabeled-unlabeled learning while the other uses a novel, recursive risk estimator. Experimental results demonstrate our methods' effectiveness across numerous real-world datasets and forms of positive bias, including disjoint positive class-conditional supports. Additionally, we propose a general, simplified approach to address PU risk estimation overfitting.