Spectral Conditions for the Ingleton Inequality

2026-02-18

AI Summary

This study investigates the extent to which the Ingleton inequality fails for entropy vectors associated with non-representable matroids exhibiting strong non-extractable mutual information. Focusing on pairs of random variables $(X,Y)$ uniformly distributed over the support set defined by an expanded graph, the authors combine spectral graph mixing lemmas with set partitioning techniques to establish, for the first time, an explicit lower bound on the Ingleton quantity in terms of spectral graph parameters. The results demonstrate that even in the presence of strong non-extractable mutual information, the Ingleton inequality remains approximately valid, deviating only by a small additive error. This reveals a quantitative influence of graph structure on the constraints imposed by information inequalities.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The Ingleton inequality is a classical linear information inequality that holds for representable matroids but fails to be universally valid for entropic vectors. Understanding the extent to which this inequality can be violated has been a longstanding problem in information theory. In this paper, we show that for a broad class of jointly distributed random variables $(X,Y)$ the Ingleton inequality holds up to a small additive error, even even though the mutual information between $X$ and $Y$ is far from being extractable. Contrary to common intuition, strongly non-extractable mutual information does not lead to large violations of the Ingleton inequality in this setting. More precisely, we consider pairs $(X,Y)$ that are uniformly distributed on their joint support and whose associated biregular bipartite graph is an expander. For all auxiliary random variables $A$ and $B$ jointly distributed with $(X,Y)$, we establish a lower bound on the Ingleton quantity $I(X:Y | A) + I(X:Y | B) + I(A:B) - I(X:Y)$ in terms of the spectral parameters of the underlying graph. Our proof combines the expander mixing lemma with a partitioning technique for finite sets.

Problem

• Ingleton inequality
• entropic vectors
• mutual information
• expander graphs
• information theory

Innovation

• Ingleton inequality
• expander graphs
• spectral parameters
• entropic vectors
• mutual information