This study introduces a new approach to addressing positive and unlabeled (PU) data through the double exponential tilting model (DETM). Traditional methods often fall short because they only apply to selected completely at random (SCAR) PU data, where the labeled positive and unlabeled positive data are assumed to be from the same distribution. In contrast, our DETM's dual structure effectively accommodates the more complex and underexplored selected at random PU data, where the labeled and unlabeled positive data can be from different distributions. We rigorously establish the theoretical foundations of DETM, including identifiability, parameter estimation, and asymptotic properties. Additionally, we move forward to statistical inference by developing a goodness-of-fit test for the SCAR condition and constructing confidence intervals for the proportion of positive instances in the target domain. We leverage an approximated Bayes classifier for classification tasks, demonstrating DETM's robust performance in prediction. Through theoretical insights and practical applications, this study highlights DETM as a comprehensive framework for addressing the challenges of PU data.

PU learning refers to the classification problem in which only part of positive samples are labeled. Existing PU learning methods treat unlabeled samples equally. However, in many real tasks, from common sense or domain knowledge, some unlabeled samples are more likely to be positive than others. In this paper, we propose soft label PU learning, in which unlabeled data are assigned soft labels according to their probabilities of being positive. Considering that the ground truth of T PR, FPR, and AUC are unknown, we then design PU counterparts of these metrics to evaluate the performances of soft label PU learning methods within validation data. We show that these new designed PU metrics are good substitutes for the real metrics. After that, a method that optimizes such metrics is proposed. Experiments on public datasets and real datasets for anti-cheat services from Tencent games demonstrate the effectiveness of our proposed method.

The goal of positive-unlabeled (PU) learning is to train a binary classifier on the basis of training data containing positive and unlabeled instances, where unlabeled observations can belong either to the positive class or to the negative class. Modeling PU data requires certain assumptions on the labeling mechanism that describes which positive observations are assigned a label. The simplest assumption, considered in early works, is SCAR (Selected Completely at Random Assumption), according to which the propensity score function, defined as the probability of assigning a label to a positive observation, is constant. On the other hand, a much more realistic assumption is SAR (Selected at Random), which states that the propensity function solely depends on the observed feature vector. SCAR-based algorithms are much simpler and computationally much faster compared to SAR-based algorithms, which usually require challenging estimation of the propensity score. In this work, we propose a relatively simple and computationally fast test that can be used to determine whether the observed data meet the SCAR assumption. Our test is based on generating artificial labels conforming to the SCAR case, which in turn allows to mimic the distribution of the test statistic under the null hypothesis of SCAR. We justify our method theoretically. In experiments, we demonstrate that the test successfully detects various deviations from SCAR scenario and at the same time it is possible to effectively control the type I error. The proposed test can be recommended as a pre-processing step to decide which final PU algorithm to choose in cases when nature of labeling mechanism is not known.

Document set expansion aims to identify relevant documents from a large collection based on a small set of documents that are on a fine-grained topic. Previous work shows that PU learning is a promising method for this task. However, some serious issues remain unresolved, i.e. typical challenges that PU methods suffer such as unknown class prior and imbalanced data, and the need for transductive experimental settings. In this paper, we propose a novel PU learning framework based on density estimation, called puDE, that can handle the above issues. The advantage of puDE is that it neither constrained to the SCAR assumption and nor require any class prior knowledge. We demonstrate the effectiveness of the proposed method using a series of real-world datasets and conclude that our method is a better alternative for the DSE task.

Positive and Unlabeled (PU) learning is a type of semi-supervised binary classification where the machine learning algorithm differentiates between a set of positive instances (labeled) and a set of both positive and negative instances (unlabeled). PU learning has broad applications in settings where confirmed negatives are unavailable or difficult to obtain, and there is value in discovering positives among the unlabeled (e.g., viable drugs among untested compounds). Most PU learning algorithms make the selected completely at random (SCAR) assumption, namely that positives are selected independently of their features. However, in many real-world applications, such as healthcare, positives are not SCAR (e.g., severe cases are more likely to be diagnosed), leading to a poor estimate of the proportion, $\alpha$, of positives among unlabeled examples and poor model calibration, resulting in an uncertain decision threshold for selecting positives. PU learning algorithms can estimate $\alpha$ or the probability of an individual unlabeled instance being positive or both. We propose two PU learning algorithms to estimate $\alpha$, calculate calibrated probabilities for PU instances, and improve classification metrics: i) PULSCAR (positive unlabeled learning selected completely at random), and ii) PULSNAR (positive unlabeled learning selected not at random). PULSNAR uses a divide-and-conquer approach that creates and solves several SCAR-like sub-problems using PULSCAR. In our experiments, PULSNAR outperformed state-of-the-art approaches on both synthetic and real-world benchmark datasets.

We demonstrate the difference The nature of anonymity and decentralization of blockchain between the estimated values of evaluation metrics and the systems are making changes in the finance industry due actual values through an engineered PU dataset from the to its immutability, transparency, and automation [1]. Such Ethereum transaction dataset proposed in [15] to show the decentralized systems, however, are in the current stage of a concerns of assuming unlabeled data to be normal. We conduct temporarily unregulated environment [2], [3] with a variety of experiments to show that applying various PU classifiers can abnormal usages and security concerns. The abnormal usages help in improving the classification performance on two realworld include both the illicit activities clearly defined by traditional datasets with limited positive labels. The PU classifiers fiance: phishing scams, Ponzi schemes, money laundering, estimate potential identifiable class prior or treat the unlabeled etc. [4], and also the activities with no clear definition of examples as negative samples with label noise and learn with lawfulness or being just defined such as mixing services, i.e., biased models. We also compare various graph representation the mixer nodes involve in funds to confuse the trace of the methods for extracting node embedding vectors as the input transfers from the original source, e.g., US Department of the to get diverse data distribution for the same data to obtain Treasury declared Tornado Cash as a sanctioned entity [5], [6].

Positive-unlabeled learning (PU learning) is known as a special case of semi-supervised binary classification where only a fraction of positive examples are labeled. The challenge is then to find the correct classifier despite this lack of information. Recently, new methodologies have been introduced to address the case where the probability of being labeled may depend on the covariates. In this paper, we are interested in establishing risk bounds for PU learning under this general assumption. In addition, we quantify the impact of label noise on PU learning compared to standard classification setting. Finally, we provide a lower bound on minimax risk proving that the upper bound is almost optimal.

Most positive and unlabeled data is subject to selection biases. The labeled examples can, for example, be selected from the positive set because they are easier to obtain or more obviously positive. This paper investigates how learning can be enabled in this setting. We propose and theoretically analyze an empirical-risk-based method for incorporating the labeling mechanism. Additionally, we investigate under which assumptions learning is possible when the labeling mechanism is not fully understood and propose a practical method to enable this. Our empirical analysis supports the theoretical results and shows that taking into account the possibility of a selection bias, even when the labeling mechanism is unknown, improves the trained classifiers.

For many interesting tasks, such as medical diagnosis and web page classification, a learner only has access to some positively labeled examples and many unlabeled examples. Learning from this type of data requires making assumptions about the true distribution of the classes and/or the mechanism that was used to select the positive examples to be labeled. The commonly made assumptions, separability of the classes and positive examples being selected completely at random, are very strong. This paper proposes a weaker assumption that assumes the positive examples to be selected at random, conditioned on some of the attributes. To learn under this assumption, an EM method is proposed. Experiments show that our method is not only very capable of learning under this assumption, but it also outperforms the state of the art for learning under the selected completely at random assumption.