Abstract
Weakly supervised learning often operates with coarse aggregate signals rather than instance labels. We study a setting where each training example is an $n$-tuple containing exactly m positives, while only the count m per tuple is observed. This NTMP (N-tuple with M positives) supervision arises in, e.g., image classification with region proposals and multi-instance measurements. We show that tuple counts admit a trainable unbiased risk estimator (URE) by linking the tuple-generation process to latent instance marginals. Starting from fixed (n,m), we derive a closed-form URE and extend it to variable tuple sizes, variable counts, and their combination. Identification holds whenever the effective mixing rate is separated from the class prior. We establish generalization bounds via Rademacher complexity and prove statistical consistency with standard rates under mild regularity assumptions. To improve finite-sample stability, we introduce simple ReLU corrections to the URE that preserve asymptotic correctness. Across benchmarks converted to NTMP tasks, the approach consistently outperforms representative weak-supervision baselines and yields favorable precision-recall and F1 trade-offs. It remains robust under class-prior imbalance and across diverse tuple configurations, demonstrating that count-only supervision can be exploited effectively through a theoretically grounded and practically stable objective.
Abstract (translated)
弱监督学习通常处理粗略的聚合信号，而不是实例标签。我们研究了一种设置，在这种设置中，每个训练样本是一个包含恰好m个正例的n元组，而我们只观察到每组中的计数m。这种NTMP（N-tuple with M positives）监督出现在例如带有区域提案和多实例测量的图像分类场景中。我们展示了元组计数可以提供一个可训练的无偏风险估计器（URE），通过将元组生成过程与潜在实例边缘联系起来。从固定的(n, m)开始，我们推导出一种封闭形式的URE，并将其扩展到变量大小、变量数量以及它们的组合。当有效的混合速率与类先验分离时，即可进行识别。我们通过Rademacher复杂度建立泛化界限，并在适度正则性假设下证明统计一致性及标准速度。为了提高有限样本稳定性，我们在URE中引入了简单的ReLU修正，以保持渐近正确性。在转换为NTMP任务的基准测试上，该方法始终优于代表性弱监督基线，在精度-召回率和F1指标权衡方面表现良好，并且即使在类先验不平衡的情况下以及面对各种元组配置时也能保持稳健性。这表明仅通过计数监督就可以有效地利用一个理论基础坚实、实践稳定性好的目标函数。
URL
https://arxiv.org/abs/2510.18406
