Revisiting the Stability of the Ingleton Inequality: A Tropicalization-Free Approach
Information Theory
Summary
The authors revisit a known result about the Ingleton inequality, which relates to information theory and holds true under certain exact conditions of independence. Previous work by Matveev and Romashchenko studied what happens when these conditions are almost but not completely met, using a complicated method called tropical probability. The authors provide a simpler approach without using tropical methods, making the proofs clearer and improving the error estimates. They also solve an open problem by finding a new set of related entropy inequalities that show how stable the sum of two Ingleton expressions is.
Authors
Laszlo Csirmaz
Abstract
The classical Ingleton inequality is known to hold for entropic points under specific exact conditional independence constraints. Recently, Matveev and Romashchenko (2026) investigated the stability of these implications, quantifying the extent to which the Ingleton inequality can be violated when a group of conditional mutual information terms is small but non-zero. While their proofs relied fundamentally on the complex framework of tropical probability spaces, we revisit these stability results using a completely tropicalization-free approach. By developing an alternative framework, we significantly streamline the underlying concepts and proofs, derive explicit error terms, and improve some estimates. Furthermore, we resolve an open problem posed in prior work by exhibiting a new infinite family of entropy inequalities that establishes the stability of the sum of two Ingleton expressions.