Spectral Conditions for the Ingleton Inequality
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911693549862912 |
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| author | Matveev, Rostislav Romashchenko, Andrei |
| author_facet | Matveev, Rostislav Romashchenko, Andrei |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_16536 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spectral Conditions for the Ingleton Inequality Matveev, Rostislav Romashchenko, Andrei Information Theory 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. |
| title | Spectral Conditions for the Ingleton Inequality |
| topic | Information Theory |
| url | https://arxiv.org/abs/2602.16536 |