用于分类性能评估的统一代数框架
A Unified Algebraic Framework for Classification Performance Evaluation
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中文总结 AI 辅助
提出统一代数框架评估分类性能，涵盖多种设置。以二值指示矩阵表示标签，结合三种聚合算子，可自动扩展二值性能度量到所有设置，还适应多种情况，建立了理论结果并经实证验证。
中文摘要 AI 辅助
我们提出了一个用于分类性能评估的统一代数框架，该框架在单一形式体系中涵盖了二分类、多分类、多标签、有序、层次、成本敏感和软标签设置。其基础是将实际标签和预测标签表示为二值指示矩阵，并结合三个聚合算子——全局、按列和按行——分别对应于微观、宏观/加权和示例平均。通过替换这些算子，任何以真/正/负计数表示的二值性能度量都能自动扩展到所有设置，无需针对特定度量进行推导即可生成多分类和多标签版本。该框架还通过argmax或阈值处理来适应软分类器输出，通过三角范数来适应软真值，通过隶属函数或累积编码来适应有序分类，以及通过包含MAE和MSE作为特殊情况的成本矩阵来适应成本敏感评估。我们建立了几个理论结果：微观平均等于分母加权宏观平均；乘积t范数是唯一保持混淆矩阵划分的范数；偏斜不变度量被表征为召回率和特异性的函数；在多分类设置中，微观精度、微观召回率和微观F1都等于准确率。在合成数据和真实数据上的实证说明证实了这些理论发现。
英文摘要
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).
发表机构
- Universidade Federal do ABC (UFABC)（巴西联邦大学ABC分校）
机构由 AI 辅助整理，请以论文原文为准。
