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

[2504.02169] On the Geometry of Receiver Operating Characteristic and Precision-Recall Curves Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Computer Science > Machine Learning arXiv:2504.02169 (cs) [Submitted on 2 Apr 2025 ( v1 ), last revised 14 Apr 2026 (this version, v3)] Title: On the Geometry of Receiver Operating Characteristic and Precision-Recall Curves Authors: Reza Sameni View a PDF of the paper titled On the Geometry of Receiver Operating Characteristic and Precision-Recall Curves, by Reza Sameni View PDF HTML (experimental) 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. Subjects: Machine Learning (cs.LG) ; Artificial Intelligence (cs.AI); Statistics Theory (math.ST); Machine Learning (stat.ML) Cite as: arXiv:2504.02169 [cs.LG] (or arXiv:2504.02169v3 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2504.02169 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Reza Sameni [ view email ] [v1] Wed, 2 Apr 2025 23:04:28 UTC (164 KB) [v2] Thu, 12 Jun 2025 15:36:27 UTC (165 KB) [v3] Tue, 14 Apr 2026 02:37:42 UTC (166 KB) Full-text links: Access Paper: View a PDF of the paper titled On the Geometry of Receiver Operating Characteristic and Precision-Recall Curves, by Reza Sameni View PDF HTML (experimental) TeX Source view license Current browse context: cs.LG < prev | next > new | recent | 2025-04 Change to browse by: cs cs.AI math math.ST stat stat.ML stat.TH References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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