用数学不等式统一约束分类器在诊断中的误差，提升结果可靠性。
Inequalities for Optimization of Classification Algorithms: A Perspective Motivated by Diagnostic Testing
- 基于集合论重构分类与流行率估计问题，用混淆矩阵变体建模
- 通过格什戈林定理证明最大半径ρ_m可统一控制两类误差上限
- 二分类下可通过测度论“水位法”优化，适合医疗诊断场景
受医学诊断典型问题启发，本文提出并研究一种目标函数，可对分类器及相关数据分析工具中感兴趣量的不确定性进行统一上界约束。首先采用集合论视角，将诊断中的两大任务——分类与流行率估计——重新表述为监督学习中通常使用的混淆矩阵${oldsymbol { m P}}$的一种变形。随后结合条件概率与格什戈林圆定理，证明矩阵$\b I - \boldsymbol { m P}$（其中$\b I$为单位矩阵）的最大格什戈林半径$oldsymbol ρ_m$能提供分类与流行率估计的统一误差上界。在二分类情形下，通过测度论的“水位法”论证，可最小化$oldsymbol ρ_m$，从而优化生成矩阵${oldsymbol { m P}}$的适当划分$U$。文章还通过一例说明二分类解难以推广至多分类情形，并推导出混淆矩阵的相关性质。
原文摘要 · Abstract (English)
Motivated by canonical problems in medical diagnostics, we propose and study properties of an objective function that uniformly bounds uncertainties in quantities of interest extracted from classifiers and related data analysis tools. We begin by adopting a set-theoretic perspective to show how two main tasks in diagnostics -- classification and prevalence estimation -- can be recast in terms of a variation on the confusion (or error) matrix ${\boldsymbol {\rm P}}$ typically considered in supervised learning. We then combine arguments from conditional probability with the Gershgorin circle theorem to demonstrate that the largest Gershgorin radius $\boldsymbol ρ_m$ of the matrix $\mathbb I-\boldsymbol {\rm P}$ (where $\mathbb I$ is the identity) yields uniform error bounds for both classification and prevalence estimation. In a two-class setting, $\boldsymbol ρ_m$ is minimized via a measure-theoretic ``water-leveling'' argument that optimizes an appropriately defined partition $U$ generating the matrix ${\boldsymbol {\rm P}}$. We also consider an example that illustrates the difficulty of generalizing the binary solution to a multi-class setting and deduce relevant properties of the confusion matrix.
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