Construct a biased SVM classifier based on Chebyshev distance for PU learning

Abstract

In some real applications, only limited labeled positive examples and many unlabeled examples are available, but there are no negative examples. Such learning is termed as positive and unlabeled (PU) learning. PU learning algorithm has been studied extensively in recent years. However, the classical ones based on the Support Vector Machines (SVMs) are assumed that labeled positive data is independent and identically distributed (i.i.d) and the sample size is large enough. It leads to two obvious shortcomings. On the one hand, the performance is not satisfactory, especially when the number of the labeled positive examples is small. On the other hand, classification results are not optimistic when datasets are Non-i.i.d. For this reason, this paper proposes a novel SVM classifier using Chebyshev distance to measure the empirical risk and designs an efficient iterative algorithm, named L∞ - BSVM in short. L∞ - BSVM includes the following merits: (1) it allows all sample points to participate in learning to prompt classification performance, especially in the case where the size of labeled data is small; (2) it minimizes the distance of the sample points that are (outliers in Non-i.i.d) farthest from the hyper-plane, where outliers are sufficiently taken into consideration (3) our iterative algorithm can solve large scale optimization problem with low time complexity and ensure the convergence of the optimum solution. Finally, extensive experiments on three types of datasets: artificial Non-i.i.d datasets, fault diagnosis of railway turnout with few labeled data (abnormal turnout) and six benchmark real-world datasets verify above opinions again and demonstrate that our classifier is much better than state-of-the-art competitors, such as B-SVM, LUHC, Pulce, B-LSSVM, NB and so on.