标题： 接收机工作特性曲线与精确率-召回率曲线的几何学研究 显示英文标题

标题： On the Geometry of Receiver Operating Characteristic and Precision-Recall Curves

摘要： 我们研究了二分类问题中接收者操作特征（ROC）曲线和精确率-召回率（PR）曲线的几何性质。关键发现是，许多最常用的二分类指标仅仅是组成函数 $G := F_p \circ F_n^{-1}$ 的函数，其中 $F_p(\cdot)$ 和 $F_n(\cdot)$ 分别是正类和负类中分类器得分的类别条件累积分布函数。 这种几何视角有助于选择工作点、理解决策阈值的影响以及比较不同分类器。它还帮助解释了ROC/PR曲线的形状和几何特性如何反映分类器的行为，并提供了构建针对特定应用和上下文约束优化的分类器的客观工具。 我们进一步探讨了分类器占优的条件，通过分析和数值例子展示了类别可分性和方差对ROC和PR几何特性的影响，并推导了正类到负类泄漏函数 $G(\cdot)$ 与Kullback-Leibler散度之间的联系。该框架强调了实际考虑因素，如模型校准、代价敏感优化以及在现实世界容量限制下的工作点选择，从而促进了更明智的分类器部署和决策方法。 显示英文摘要

摘要： We study the geometry of Receiver Operating Characteristic (ROC) and Precision-Recall (PR) curves in binary classification problems. The key finding is that many of the most commonly used binary classification metrics are merely functions of the composition function $G := F_p \circ F_n^{-1}$, where $F_p(\cdot)$ and $F_n(\cdot)$ are the class-conditional cumulative distribution functions of the classifier scores in the positive and negative classes, respectively. This geometric perspective facilitates the selection of operating points, understanding the effect of decision thresholds, and comparison between classifiers. It also helps explain how the shapes and geometry of ROC/PR curves reflect classifier behavior, providing objective tools for building classifiers optimized for specific applications with context-specific constraints. We further explore the conditions for classifier dominance, present analytical and numerical examples demonstrating the effects of class separability and variance on ROC and PR geometries, and derive a link between the positive-to-negative class leakage function $G(\cdot)$ and the Kullback--Leibler divergence. The framework highlights practical considerations, such as model calibration, cost-sensitive optimization, and operating point selection under real-world capacity constraints, enabling more informed approaches to classifier deployment and decision-making.