🤖 AI Summary
This work investigates the geometric foundations of ROC and PR curves in binary classification, aiming to unify the understanding of curve morphology and classifier behavior through a geometric lens. Methodologically, it introduces the composite function (G = F_p circ F_n^{-1}) as a core modeling framework—where (F_p) and (F_n) denote the CDFs of positive and negative class score distributions—and rigorously establishes a geometric mapping between ROC/PR curve shapes and the underlying distributional geometry. It reveals that (G) quantifies inter-class leakage and admits interpretation via KL divergence. Furthermore, it derives geometric criteria for classifier dominance and interpretability grounded in differential geometry, statistical inference, and CDF transformation theory. The contributions include: (i) a principled, geometrically interpretable framework for threshold selection; (ii) robust, distribution-agnostic tools for classifier comparison; and (iii) enhanced reliability and adaptability in cost-sensitive deployment—particularly under class imbalance and distributional overlap.
📝 Abstract
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.