On a Construction of Entropic Vectors Using Lattice-Generated Distributions

Hassibi, Babak; Shadbakht, S.

doi:10.1109/isit.2007.4557096

Cited by 18 publications

(11 citation statements)

References 17 publications

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“…Despite this characterization, even Γ * 4 is still not fully understand. Since then, many authors has been investigating the properties of Γ * N with the hope of ultimately fully characterizing the region [6,[9][10][11][12][13][14].…”

Section: The Region Of Entropic Vectorsmentioning

confidence: 99%

“…Indeed, the subspace rank function vector is merely formed by taking some of the elements from the 2 N -1 representable matroid rank function vector associated with G. That is, rank function vectors created via (9) are projections of rank function vectors of representable matroids. Rank functions capable of being represented in the manner for some N , q and G, are called subspace ranks in some contexts [16][17][18], while other papers effectively define a collection of vector random variables created in this manner a subspace arrangement [19].…”

Section: Structure Of γmentioning

confidence: 99%

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Only One Nonlinear Non-Shannon Inequality is Necessary for Four Variables

Liu

Walsh

2015

Proceedings of 2nd International Electronic Conference on Entropy and Its Applications

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The region of entropic vectors Γ * N has been shown to be at the core of determining fundamental limits for network coding, distributed storage, conditional independence relations, and information theory. Characterizing this region is a problem that lies at the intersection of probability theory, group theory, and convex optimization. A 2 N -1 dimensional vector is said to be entropic if each of its entries can be regarded as the joint entropy of a particular subset of N discrete random variables. While the explicit characterization of the region of entropic vectors Γ * N is unknown for N 4, here we prove that only one form of nonlinear non-shannon inequality is necessary to fully characterize Γ * 4 . We identify this inequality in terms of a function that is the solution to an optimization problem. We also give some symmetry and convexity properties of this function which rely on the structure of the region of entropic vectors and Ingleton inequalities. This result shows that inner and outer bounds to the region of entropic vectors can be created by upper and lower bounding the function that is the answer to this optimization problem.

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Section: The Region Of Entropic Vectorsmentioning

confidence: 99%

Section: Structure Of γmentioning

confidence: 99%

Only One Nonlinear Non-Shannon Inequality is Necessary for Four Variables

Liu

Walsh

2015

Proceedings of 2nd International Electronic Conference on Entropy and Its Applications

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“…5 shows an example of a lattice. We define a probability distribution on this lattice as follows [18], Definition 1 (Lattice-Generated Distribution): A probability distribution over n random variables, each with alphabet size N , is called lattice-generated, if for some lattice L(M ),…”

Section: B Entropy Region -Discrete Random Variablesmentioning

confidence: 99%

“…Enforcing the quasi-uniformity on the defined distribution gives the following Lemma [18], Lemma 1 (Lattice-Generated Quasi-Uniform Distributions): A lattice-generated distribution is quasi-uniform if the lattice has a period that divides N . The latter is true if, and only if, the matrix M −1 N has integer entries.…”

Section: B Entropy Region -Discrete Random Variablesmentioning

confidence: 99%

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On the entropy region of discrete and continuous random variables and network information theory

Shadbakht

Hassibi

2008

2008 42nd Asilomar Conference on Signals, Systems and Computers

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Abstract-We show that a large class of network information theory problems can be cast as convex optimization over the convex space of entropy vectors. A vector in 2 n − 1 dimensional space is called entropic if each of its entries can be regarded as the joint entropy of a particular subset of n random variables (note that any set of size n has 2 n − 1 nonempty subsets.) While an explicit characterization of the space of entropy vectors is wellknown for n = 2, 3 random variables, it is unknown for n > 3 (which is why most network information theory problems are open.) We will construct inner bounds to the space of entropic vectors using tools such as quasi-uniform distributions, lattices, and Cayley's hyperdeterminant.

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A recursive construction of the set of binary entropy vectors

Walsh

Weber

2009

2009 47th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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The primary contribution is a finite terminating algorithm that determines membership of a candidate entropy vector in the set of binary entropy vectors Φ N . We outline the relationship between Φ N and its unbounded cardinality discrete random variable counterpartΓ * N (or its normalizationΩ * N ). We discuss connections between Φ N andΩ * N . For example, for any outer bound, say the Shannon outer bound P N , toΩ * N , we provide a finite terminating algorithm to find a polytopic inner bound on Ω * N that agrees on tight faces of the outer bound.

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On a Construction of Entropic Vectors Using Lattice-Generated Distributions

Cited by 18 publications

References 17 publications

Only One Nonlinear Non-Shannon Inequality is Necessary for Four Variables

Only One Nonlinear Non-Shannon Inequality is Necessary for Four Variables

On the entropy region of discrete and continuous random variables and network information theory

A recursive construction of the set of binary entropy vectors

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