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Spectral Conditions for the Ingleton Inequality

Rostislav Matveev, Andrei Romashchenko

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.

Spectral Conditions for the Ingleton Inequality

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 the Ingleton inequality holds up to a small additive error, even even though the mutual information between and 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 that are uniformly distributed on their joint support and whose associated biregular bipartite graph is an expander. For all auxiliary random variables and jointly distributed with , we establish a lower bound on the Ingleton quantity in terms of the spectral parameters of the underlying graph. Our proof combines the expander mixing lemma with a partitioning technique for finite sets.
Paper Structure (29 sections, 22 theorems, 103 equations)

This paper contains 29 sections, 22 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Proof technique.
  3. Preliminaries
  4. Notations
  5. Random Variables
  6. Graphs
  7. Case 1: $\bm {t\leq s}$.
  8. Case 2: $\bm {t>s}$.
  9. Graphs and random variables
  10. Ingleton bounds for quasi-uniform tuples
  11. Uniform tuples
  12. Case 1: At least one of the following two inequalities hold.
  13. Case 2: Both of the following two inequalities hold.
  14. Quasi-uniform tuples
  15. Case 1: At least one of the following two inequalities hold.
  16. ...and 14 more sections

Key Result

Theorem 1

Let $\varepsilon_{0}$ be a positive constant. Suppose $(X,Y)$ is ob-tain-ed by choosing a uniformly random edge in a biregular bipartite graph $G=(\Xsf\sqcup\Ysf,\Esf)$ whose largest and second-largest eigenvalues are $\lambda_1$ and $\lambda_2$, respectively. Assume that graph $G$ is such that $I(X where the $O(\cdot)$-summand implicitly depends on the chosen threshold value $\varepsilon_{0}$.

Theorems & Definitions (41)

  • Theorem 1: Main result, see Theorem \ref{['p:main']} in Section \ref{['s:main']}
  • remark 1
  • remark 2
  • Example 2: See Section \ref{['s:linear']}
  • Example 3: See Section \ref{['s:algebraic']}
  • remark 3
  • lemma 2.2.1
  • lemma 2.2.2
  • proof
  • lemma 2.2.3
  • ...and 31 more