Cayley's hyperdeterminant, the principal minors of a symmetric matrix and the entropy region of 4 Gaussian random variables

Shadbakht, S.; Hassibi, Babak

doi:10.1109/allerton.2008.4797553

Cited by 7 publications

(14 citation statements)

References 13 publications

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“…Moreover, No. 8', 9', 13' and 15' correspond to Instances 8,9,13 and 15 when p = 2 but q−1 2 is odd, in which case Ingleton is satisfied. Finally, the order calculation for Instances 12-15 only works for p = 3.…”

Section: Ingleton Violations In Gl(2 Q)mentioning

confidence: 99%

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On Ingleton-Violating Finite Groups

Mao

Thill

Hassibi

2017

IEEE Trans. Inform. Theory

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Given n discrete random variables, its entropy vector is the 2 n − 1 dimensional vector obtained from the joint entropies of all non-empty subsets of the random variables. It is well known that there is a one-toone correspondence between such an entropy vector and a certain group-characterizable vector obtained from a finite group and n of its subgroups [3]. This correspondence may be useful for characterizing the space of entropic vectors and for designing network codes. If one restricts attention to abelian groups then not all entropy vectors can be obtained. This is an explanation for the fact shown by Dougherty et al[4] that linear network codes cannot achieve capacity in general network coding problems (since linear network codes form an abelian group). All abelian group-characterizable vectors, and by fiat all entropy vectors generated by linear network codes, satisfy a linear inequality called the Ingleton inequality. It is therefore of interest to identify groups that violate the Ingleton inequality. In this paper, we study the problem of finding nonabelian finite groups that yield characterizable vectors which violate the Ingleton inequality. Using a refined computer search, we find the symmetric group S 5 to be the smallest group that violates the Ingleton inequality. Careful study of the structure of this group, and its subgroups, reveals that it belongs to the Ingleton-violating family P GL(2, q) with a prime power q ≥ 5, i.e., the projective group of 2 × 2 nonsingular matrices with entries in F q . We further interpret this family of groups, and their subgroups, using the theory of group actions and identify the subgroups as certain stabilizers. We also extend the construction to more general groups such as P GL(n, q) and GL(n, q). The families of groups identified here are therefore good candidates for constructing network codes more powerful than linear network codes, and we discuss some considerations for constructing such group network codes. Index TermsPortions of this work were presented at the

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Section: Ingleton Violations In Gl(2 Q)mentioning

confidence: 99%

“…In fact, in the original paper of Chan and Yeung[3] the same type of subgroups are also used in to show that every entropy vector can be approximated by a scaled group-characterizable vector 9. violating subgroups are identified, in Section VI.…”

mentioning

confidence: 99%

On Ingleton-Violating Finite Groups

Mao

Thill

Hassibi

2017

IEEE Trans. Inform. Theory

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“…Despite this importance, the entropy region Γ * n is only known for n = 2, 3 random variables and remains unknown for n ≥ 4 random variables. Nonetheless, there are important connections known between Γ * n and matroid theory (since entropy is a submodular function and therefore somehow defines a matroid) [5], determinantal inequalities (through the connection with Gaussian random variables) [6], and quasi-uniform arrays [7]. However, perhaps most intriguing is the connection to finite groups which we briefly elaborate below.…”

Section: Introductionmentioning

confidence: 99%

Violating the Ingleton inequality with finite groups

Mao¹,

Hassibi²

2009

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

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It is well known that there is a one-to-one correspondence between the entropy vector of a collection of n random variables and a certain group-characterizable vector obtained from a finite group and n of its subgroups [1]. However, if one restricts attention to abelian groups then not all entropy vectors can be obtained. This is an explanation for the fact shown by Dougherty et al [2] that linear network codes cannot achieve capacity in general network coding problems (since linear network codes form an abelian group). All abelian group-characterizable vectors, and by fiat all entropy vectors generated by linear network codes, satisfy a linear inequality called the Ingleton inequality. In this paper, we study the problem of finding nonabelian finite groups that yield characterizable vectors which violate the Ingleton inequality. Using a refined computer search, we find the symmetric group S 5 to be the smallest group that violates the Ingleton inequality. Careful study of the structure of this group, and its subgroups, reveals that it belongs to the Ingleton-violating family P GL(2, p) with primes p ≥ 5, i.e., the projective group of 2 × 2 nonsingular matrices with entries in F p . This family of groups is therefore a good candidate for constructing network codes more powerful than linear network codes.

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“…T . From (24) it is obvious that studying the entropy of Gaussian random variables requires the study of determinantal inequalities and the relations between the principal minors of a positive definite symmetric matrix [21]. One of the important relations as it turns out is the socalled Cayley's hyperdeterminant [22].…”

Section: Theorem 5 (Discrete and Continuous Entropies)mentioning

confidence: 99%

“…For n ≥ 4 this characterization is under investigation where a closely related problem in characterizing the entropy region is the study of necessary and sufficient conditions for a 2 n − 1 dimensional vector to correspond to all of the principal minors of a symmetric matrix [22], [24].…”

Section: Theorem 6 (Region Of 3 Scalar Gaussian Variables)mentioning

confidence: 99%

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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Cayley's hyperdeterminant, the principal minors of a symmetric matrix and the entropy region of 4 Gaussian random variables

Cited by 7 publications

References 13 publications

On Ingleton-Violating Finite Groups

On Ingleton-Violating Finite Groups

Violating the Ingleton inequality with finite groups

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

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