Inequalities for Optimization of Classification Algorithms: A Perspective Motivated by Diagnostic Testing

TL;DR

This paper introduces a novel objective function based on the Gershgorin circle theorem to provide uniform error bounds for classification and prevalence estimation, motivated by medical diagnostics.

Contribution

It presents a set-theoretic approach and a measure-theoretic optimization to minimize the Gershgorin radius for improved error bounds in classifiers.

Findings

Gershgorin radius bounds errors in classification and prevalence estimation.

Optimal partitioning minimizes the Gershgorin radius in binary classification.

Multi-class extension presents additional challenges and properties.

Abstract

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 $\boldsymbol \rho_m$ 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, $\boldsymbol \rho_m$ 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.