IoU is not submodular

TL;DR

This paper demonstrates that the Intersection over Union (IoU), a common metric in machine learning, is not a submodular function, correcting a misconception that has influenced subsequent research.

Contribution

It clarifies a fundamental property of IoU, showing it is not submodular, which impacts how it should be used and understood in machine learning applications.

Findings

IoU is not submodular

A common misconception in literature is corrected

Implications for machine learning metrics

Abstract

This short article aims at demonstrate that the Intersection over Union (or Jaccard index) is not a submodular function. This mistake has been made in an article which is cited and used as a foundation in another article. The Intersection of Union is widely used in machine learning as a cost function especially for imbalance data and semantic segmentation.