A Unified Algebraic Framework for Classification Performance Evaluation

[2607.04028] A Unified Algebraic Framework for Classification Performance Evaluation Skip to main content Search Submit Donate Log in Computer Science > Machine Learning arXiv:2607.04028 (cs) [Submitted on 4 Jul 2026 ( v1 ), last revised 25 Aug 2026 (this version, v2)] Title: A Unified Algebraic Framework for Classification Performance Evaluation Authors: Ronaldo C. Prati View PDF HTML (experimental) Abstract: We propose a unified algebraic framework for classification performance evaluation covering binary, multiclass, multilabel, ordinal, hierarchical, cost-sensitive, and soft-label settings. Actual and predicted labels are represented as binary indicator matrices, where three aggregation operators (global, column-wise, row-wise) correspond directly to micro, macro/weighted, and exemplar averaging. Any binary measure expressed in terms of the four confusion-matrix counts extends to all these settings by substituting an operator, with no measure-specific derivation. We show that structural properties governing an extension are derivable from the binary formula. Micro-averaging equals denominator-weighted macro-averaging precisely for aggregation-decomposable (linear-fractional) measures, a strict class characterised algebraically. For soft ground truth, we prove from t-norm axioms alone that the product t-norm is the unique choice whose confusion counts preserve marginal memberships. In multiclass settings, micro-precision, micro-recall, and micro-F1 collapse identically onto accuracy. Furthermore, binary skew-invariance transfers unconditionally to multilabel aggregation, but only partially to multiclass problems. For measures with a linear numerator and prediction-independent denominator, the optimal decision threshold is the share of the numerator weight favouring a negative prediction, revealing when standard training targets the measure. Under one-hot multiclass encoding, a measure fails to attain its theoretical minimum whenever its zero-true-positive value still depends on true negatives, establishing non-trivial performance floors even for completely incorrect classifiers (e.g., zero correct predictions on 10 classes yields a label accuracy of 0.8). Subjects: Machine Learning (cs.LG) ; Artificial Intelligence (cs.AI) ACM classes: I.2.6 Cite as: arXiv:2607.04028 [cs.LG] (or arXiv:2607.04028v2 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2607.04028 Focus to learn more arXiv-issued DOI via DataCite Related DOI : https://doi.org/10.1016/j.inffus.2026.104767 Focus to learn more DOI(s) linking to related resources Submission history From: Ronaldo Prati [ view email ] [v1] Sat, 4 Jul 2026 21:00:29 UTC (114 KB) [v2] Tue, 25 Aug 2026 01:15:59 UTC (221 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Additional Features Audio Summary Current browse context: cs.LG < prev | next > new | recent | 2026-07 Change to browse by: cs cs.AI References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) IArxiv recommender toggle IArxiv Recommender ( What is IArxiv? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. 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