Classification with positive and unlabeled (PU) data frequently arises in bioinformatics, clinical data, and ecological studies, where collecting negative samples can be prohibitively expensive. While prior works on PU data focus on binary classification, in this paper we consider multiple positive labels, a practically important and common setting. We introduce a multinomial-PU model and an ordinal-PU model, suited to unordered and ordered labels respectively. We propose proximal gradient descent-based algorithms to minimize the l_{1,2}-penalized log-likelihood losses, with convergence guarantees to stationary points of the non-convex objective. Despite the challenging non-convexity induced by the presence-only data and multi-class labels, we prove statistical error bounds for the stationary points within a neighborhood around the true parameters under the high-dimensional regime. This is made possible through a careful characterization of the landscape of the log-likelihood loss in the neighborhood. In addition, simulations and two real data experiments demonstrate the empirical benefits of our algorithms compared to the baseline methods.

翻译：正类与未标注（PU）数据分类问题在生物信息学、临床数据和生态研究中频繁出现，这些领域收集负类样本的成本往往过高。现有关于PU数据的研究主要聚焦于二分类问题，而本文考虑一种具有重要实际意义的常见场景——多类正标签。我们分别提出适用于无序标签和有序标签的多项式PU模型与序数PU模型。基于近端梯度下降法，我们设计了算法以最小化带l_{1,2}惩罚的对数似然损失函数，并保证其收敛至非凸目标函数的驻点。尽管仅有正标签数据与多类标签导致了极具挑战性的非凸性，我们仍证明了在高维框架下，当驻点位于真实参数邻域内时，其统计误差界的存在性。这一结果得益于对邻域内对数似然损失函数地貌的精细刻画。此外，模拟实验与两项真实数据实验表明，相较于基线方法，我们的算法具有显著的实证优势。