We present a detailed study of cardinality-aware top-$k$ classification, a novel approach that aims to learn an accurate top-$k$ set predictor while maintaining a low cardinality. We introduce a new target loss function tailored to this setting that accounts for both the classification error and the cardinality of the set predicted. To optimize this loss function, we propose two families of surrogate losses: cost-sensitive comp-sum losses and cost-sensitive constrained losses. Minimizing these loss functions leads to new cardinality-aware algorithms that we describe in detail in the case of both top-$k$ and threshold-based classifiers. We establish $H$-consistency bounds for our cardinality-aware surrogate loss functions, thereby providing a strong theoretical foundation for our algorithms. We report the results of extensive experiments on CIFAR-10, CIFAR-100, ImageNet, and SVHN datasets demonstrating the effectiveness and benefits of our cardinality-aware algorithms.

翻译：本文对基数感知的Top-$k$分类进行了详细研究，这是一种旨在学习准确Top-$k$集合预测器同时保持低基数的新方法。我们为此场景量身定制了一种新的目标损失函数，该函数同时考虑了分类误差和预测集合的基数。为优化此损失函数，我们提出了两类替代损失函数：成本敏感型求和损失与成本敏感型约束损失。最小化这些损失函数可推导出新的基数感知算法，我们针对Top-$k$分类器和基于阈值的分类器均进行了详细阐述。我们为基数感知替代损失函数建立了$H$-一致性边界，从而为算法提供了坚实的理论基础。我们在CIFAR-10、CIFAR-100、ImageNet和SVHN数据集上进行了大量实验，结果证明了基数感知算法的有效性和优势。