Extended formulations, nonnegative factorizations, and randomized communication protocols

Faenza, Yuri; Fiorini, Samuel; Grappe, Roland; Tiwary, Hans Raj

doi:10.1007/s10107-014-0755-3

Cited by 34 publications

(50 citation statements)

References 31 publications

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“…The intuition behind this is that communication complexity often deals with boolean functions where the complexity mainly stems from the combinatorial structure. In contrast, in the framework in [5] and [6] arbitrary nonnegative values can be output, potentially containing a lot of information, and as the protocol has to be only correct in expectation, this model breaks several tradeoffs in communication (e.g., probability of being correct and bits exchanged are typically strongly related). We overcome these limitations by departing from the classical views of nonnegative matrix factorization as communication problems and related concepts (see e.g., [7] and [8]).…”

Section: Introductionmentioning

confidence: 89%

“…Despite its many applications in different disciplines, our analysis is conducted with approximate extended formulations in mind. In fact due to Yannakakis's factorization theorem (see [3], [4]) and the equivalence to a communication model given in [5] and [6], it turns out that many open problems regarding the size of an optimal (exact or approximate) linear representation of a combinatorial optimization problem are equivalent to questions about the nonnegative rank of certain matrices and the related communication problems. These open problems include such important questions as to whether the matching polytope admits a linear programming formulation of polynomial size, whether MAXCUT cannot be approximated better than 1/2 − ε via any linear program of polynomial size, or whether a generic polygon needs a linear number of inequalities in any linear representation in the worst case.…”

Section: Introductionmentioning

confidence: 99%

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Common Information and Unique Disjointness

Braun,

Pokutta

2013

2013 IEEE 54th Annual Symposium on Foundations of Computer Science

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We provide a new framework for establishing strong lower bounds on the nonnegative rank of matrices by means of common information, a notion previously introduced in [1]. Common information is a natural lower bound for the nonnegative rank of a matrix and by combining it with Hellinger distance estimations we can compute the (almost) exact common information of UDISJ partial matrix. We also establish robustness of this estimation under various perturbations of the UDISJ partial matrix, where rows and columns are randomly or adversarially removed or where entries are randomly or adversarially altered. This robustness translates, via a variant of Yannakakis' Factorization Theorem, to lower bounds on the average case and adversarial approximate extension complexity. We present the first family of polytopes, the hard pair introduced in [2] related to the CLIQUE problem, with high average case and adversarial approximate extension complexity. We also provide an information theoretic variant of the fooling set method that allows us to extend fooling set lower bounds from extension complexity to approximate extension complexity.

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Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Common Information and Unique Disjointness

Braun,

Pokutta

2013

2013 IEEE 54th Annual Symposium on Foundations of Computer Science

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“…Faenza et al [5] proved equivalence of extended formulations to communication complexity. Recently there has been also significant progress in terms of lower bounding the linear extension complexity of polytopes by means of information theory, see Braverman and Moitra [4] and Braun and Pokutta [2].…”

Section: Does Every 0/1 Polytope Have An Efficient Semidefinite Lift?mentioning

confidence: 99%

On the existence of 0/1 polytopes with high semidefinite extension complexity

2014

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In Rothvoß (Math Program 142(1-2):255-268, 2013) it was shown that there exists a 0/1 polytope (a polytope whose vertices are in {0, 1} n ) such that any higher-dimensional polytope projecting to it must have 2 Ω(n) facets, i.e., its linear extension complexity is exponential. The question whether there exists a 0/1 polytope with high positive semidefinite extension complexity was left open. We answer this question in the affirmative by showing that there is a 0/1 polytope such that any spectrahedron projecting to it must be the intersection of a semidefinite cone of dimension 2 Ω(n) and an affine space. Our proof relies on a new technique to rescale semidefinite factorizations.

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“…It is worth mentioning that stronger bounds on d can be placed by using generalizations of the rank [17]. Recent literature has established an intimate relation between the non-negative rank and the classical dimension and the positive semidefinite rank and the quantum dimension in similar scenarios [17,18]. It is immediate to check that the nonnegative rank of P and the positive semidefinite rank of P are lower bounds for d in the classical and quantum case respectively in the scenario considered here.…”

Section: A Dimension Estimatesmentioning

confidence: 86%

Shared randomness and device-independent dimension witnessing

Vicente

2017

Phys. Rev. A

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It has been shown that the conditional probability distributions obtained by performing measurements on an uncharacterized physical system can be used to infer its underlying dimension in a device-independent way both in the classical and quantum setting. We analyze several aspects of the structure of the sets of probability distributions corresponding to a certain dimension taking into account whether shared randomness is available as a resource or not. We first consider the so-called prepare-and-measure scenario. We show that quantumness and shared randomness are not comparable resources. That is, on the one hand, there exist behaviours that require a quantum system of arbitrarily large dimension in order to be observed while they can be reproduced with a classical physical system of minimal dimension together with shared randomness. On the other hand, there exist behaviours which require exponentially larger dimensions classically than quantumly even if the former is supplemented with shared randomness. We also show that in the absence of shared randomness, the sets corresponding to a sufficiently small dimension are negligible (zero-measure and nowhere dense) both classically and quantumly. This is in sharp contrast to the situation in which this resource is available and explains the exceptional robustness of dimension witnesses in the setting in which devices can be taken to be uncorrelated. We finally consider the Bell scenario in the absence of shared randomness and prove some non-convexity and negligibility properties of these sets for sufficiently small dimensions. This shows again the enormous difference induced by the availability or not of this resource.

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Extended formulations, nonnegative factorizations, and randomized communication protocols

Cited by 34 publications

References 31 publications

Common Information and Unique Disjointness

Common Information and Unique Disjointness

On the existence of 0/1 polytopes with high semidefinite extension complexity

Shared randomness and device-independent dimension witnessing

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