Conditional Information Inequalities for Entropic and Almost Entropic Points

Kaced, Tarik; Romashchenko, Andrei

doi:10.1109/tit.2013.2274614

Cited by 22 publications

(33 citation statements)

References 28 publications

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“…At the same time, [32, Theorem 4.1] implies that no point of ri(E ij ) is entropic. This phenomenon can be equivalently rephrased in terms of conditional information inequalities, studied recently in [23,24,25].…”

Section: Entropy Region Of Four Variablesmentioning

confidence: 99%

Entropy Region and Convolution

Matúš

Csirmaz

2016

IEEE Trans. Inform. Theory

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Abstract. The entropy region is constructed from vectors of random variables by collecting Shannon entropies of all subvectors. Its shape is studied here by means of polymatroidal constructions, notably by convolution. The closure of the region is decomposed into the direct sum of tight and modular parts, reducing the study to the tight part. The relative interior of the reduction belongs to the entropy region. Behavior of the decomposition under selfadhesivity is clarified. Results are specialized to and completed for the region of four random variables. This and computer experiments help to visualize approximations of a symmetrized part of the entropy region. Four-atom conjecture on the minimization of Ingleton score is refuted.

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Section: Entropy Region Of Four Variablesmentioning

confidence: 99%

Entropy Region and Convolution

Matúš

Csirmaz

2016

IEEE Trans. Inform. Theory

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“…A noteworthy fact is that this result cannot be obtained as a direct implication of any unconditional linear inequality for Shannon's entropy. More precisely, whatever pair of reals λ 1 , λ 2 we take, the inequality does not hold for some distribution, see [24]. We claim that Ingleton's inequality holds also under condition (2), which is weaker than (4).…”

Section: A Generalization Of a Conditional Inequality Frommentioning

confidence: 80%

“…These are the conditional linear information inequalities, which hold only for distributions that satisfy some constraints. The first nontrivial example of a conditional linear information inequality was proven in the seminal paper [6]; see a survey of other similar results in [24]. Until now, these inequalities looked like artifacts without practical or theoretical application.…”

Section: Introductionmentioning

confidence: 97%

A Conditional Information Inequality and Its Combinatorial Applications

Kaced

Romashchenko

Vereshchagin

2018

IEEE Trans. Inform. Theory

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We show that the inequality H(A|B, X) + H(A|B, Y ) H(A|B) for jointly distributed random variables A, B, X, Y , which does not hold in general case, holds under some natural condition on the support of the probability distribution of A, B, X, Y . This result generalizes a version of the conditional Ingleton inequality: if for some distribution I(X :We present two applications of our result. The first one is the following easy-to-formulate theorem on edge colorings of bipartite graphs: assume that the edges of a bipartite graph are colored in K colors so that each two edges sharing a vertex have different colors and for each pair (left vertex x, right vertex y) there is at most one color a such both x and y are incident to edges with color a; assume further that the degree of each left vertex is at least L and the degree of each right vertex is at least R. Then K LR. The second application is a new method to prove lower bounds for biclique cover of bipartite graphs. KeywordsShannon entropy, conditional information inequalities, non Shannon type information inequalities, biclique cover, edge coloring

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“…Proposition 5.6 (Double Markov Property [52]): For any X, Y, and Z with 0 ¼ IðX^ZjYÞ ¼ IðY^ZjXÞ, the maximum common function U of X and Y satisfies IðX; Y^ZjUÞ ¼ 0.…”

Section: Data Processing Inequalitymentioning

confidence: 99%

Multivariate Mutual Information Inspired by Secret-Key Agreement

Chan¹,

Al-Bashabsheh²,

Ebrahimi³

et al. 2015

Proc. IEEE

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179

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This paper motivates the multiterminal secret-key capacity as a useful measure of multivariate mutual information, develops information-theoretic properties of this measure, and makes comparisons with other existing multivariate correlation measures.ABSTRACT | The capacity for multiterminal secret-key agreement inspires a natural generalization of Shannon's mutual information from two random variables to multiple random variables. Under a general source model without helpers, the capacity is shown to be equal to the normalized divergence from the joint distribution of the random sources to the product of marginal distributions minimized over partitions of the random sources. The mathematical underpinnings are the works on co-intersecting submodular functions and the principle lattices of partitions of the Dilworth truncation. We clarify the connection to these works and enrich them with information-theoretic interpretations and properties that are useful in solving other related problems in information theory as well as machine learning.

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Conditional Information Inequalities for Entropic and Almost Entropic Points

Cited by 22 publications

References 28 publications

Entropy Region and Convolution

Entropy Region and Convolution

A Conditional Information Inequality and Its Combinatorial Applications

Multivariate Mutual Information Inspired by Secret-Key Agreement

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