A Unified Algebraic Framework for Classification Performance Evaluation
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).