Learning from M-Tuple Dominant Positive and Unlabeled Data

Label proportion learning (LLP) addresses the classification problem where multiple instances are grouped into bags, and each bag contains information about the proportion of each class. However, in practical applications, obtaining precise supervisory information regarding the proportion of instances in a specific class is challenging. To better align with real-world application scenarios and effectively leverage the proportional constraints of instances within tuples, this article proposes a generalized learning framework, M-tuple dominant positive and unlabeled (MDPU). Specifically, we first mathematically model the distribution of instances within tuples of arbitrary size, under the constraint that the number of positive instances is no fewer than that of negative instances. Then, we derive an unbiased risk estimator (URE) that satisfies risk consistency based on the empirical risk minimization (ERM) method. To mitigate the inevitable overfitting issue during training, a risk correction method is introduced, leading to the development of a corrected risk estimator. The generalization error bounds of the URE theoretically demonstrate the consistency of the proposed method. Extensive experiments on multiple datasets and comparisons with other relevant baseline methods comprehensively validate the effectiveness of the proposed learning framework.