Grothendieck inequality

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In mathematics, the Grothendieck inequality states that there is a universal constant with the following property. If Mij is an n × n (real or complex) matrix with

for all (real or complex) numbers si, tj of absolute value at most 1, then

for all vectors Si, Tj in the unit ball B(H) of a (real or complex) Hilbert space H, the constant being independent of n. For a fixed Hilbert space of dimension d, the smallest constant that satisfies this property for all n × n matrices is called a Grothendieck constant and denoted . In fact, there are two Grothendieck constants, and , depending on whether one works with real or complex numbers, respectively.[1]

The Grothendieck inequality and Grothendieck constants are named after Alexander Grothendieck, who proved the existence of the constants in a paper published in 1953.[2]

Motivation and the operator formulation[edit]

Let be an matrix. Then defines a linear operator between the normed spaces and for . The -norm of is the quantity

If , we denote the norm by .

One can consider the following question: For what value of and is maximized? Since is linear, then it suffices to consider such that contains as many points as possible, and also such that is as large as possible. By comparing for , one sees that for all .

One way to compute is by solving the following quadratic integer program:

To see this, note that , and taking the maximum over gives . Then taking the maximum over gives by the convexity of and by the triangle inequality. This quadratic integer program can be relaxed to the following semidefinite program:

It is known that exactly computing for is NP-hard, while exacting computing is NP-hard for .

One can then ask the following natural question: How well does an optimal solution to the semidefinite program approximate ? The Grothendieck inequality provides an answer to this question: There exists a fixed constant such that, for any , for any matrix , and for any Hilbert space ,

Bounds on the constants[edit]

The sequences and are easily seen to be increasing, and Grothendieck's result states that they are bounded,[2][3] so they have limits.

Grothendieck proved that where is defined to be .[4]

Krivine (1979)[5] improved the result by proving that , conjecturing that the upper bound is tight. However, this conjecture was disproved by Braverman et al. (2011).[6]

Grothendieck constant of order d[edit]

Boris Tsirelson showed that the Grothendieck constants play an essential role in the problem of quantum nonlocality: the Tsirelson bound of any full correlation bipartite Bell inequality for a quantum system of dimension d is upperbounded by .[7][8]

Lower bounds[edit]

Some historical data on best known lower bounds of is summarized in the following table.

d Grothendieck, 1953[2] Krivine, 1979[5] Davie, 1984[9] Fishburn et al., 1994[10] Vértesi, 2008[11] Briët et al., 2011[12] Hua et al., 2015[13] Diviánszky et al., 2017[14]
2 ≈ 1.41421
3 1.41724 1.41758 1.4359
4 1.44521 1.44566 1.4841
5 ≈ 1.42857 1.46007 1.46112
6 1.47017
7 1.46286 1.47583
8 1.47586 1.47972
9 1.48608
...
≈ 1.57079 1.67696

Upper bounds[edit]

Some historical data on best known upper bounds of :

d Grothendieck, 1953[2] Rietz, 1974[15] Krivine, 1979[5] Braverman et al., 2011[6] Hirsch et al., 2016[16]
2 ≈ 1.41421
3 1.5163 1.4644
4 ≈ 1.5708
...
8 1.6641
...
≈ 2.30130 2.261 ≈ 1.78221

Applications[edit]

Cut norm estimation[edit]

Given an real matrix , the cut norm of is defined by

The notion of cut norm is essential in designing efficient approximation algorithms for dense graphs and matrices. More generally, the definition of cut norm can be generalized for symmetric measurable functions so that the cut norm of is defined by

This generalized definition of cut norm is crucial in the study of the space of graphons, and the two definitions of cut norm can be linked via the adjacency matrix of a graph.

An application of the Grothendieck inequality is to give an efficient algorithm for approximating the cut norm of a given real matrix ; specifically, given an real matrix, one can find a number such that

where is an absolute constant.[17] This approximation algorithm uses semidefinite programming.

We give a sketch of this approximation algorithm. Let be matrix defined by

One can verify that by observing, if form a maximizer for the cut norm of , then

form a maximizer for the cut norm of . Next, one can verify that , where

[18]

Although not important in this proof, can be interpreted to be the norm of when viewed as a linear operator from to .

Now it suffices to design an efficient algorithm for approximating . We consider the following semidefinite program:

Then . The Grothedieck inequality implies that . Many algorithms (such as interior-point methods, first-order methods, the bundle method, the augmented Lagrangian method) are known to output the value of a semidefinite program up to an additive error  in time that is polynomial in the program description size and .[19] Therefore, one can output which satisfies

Szemerédi's regularity lemma[edit]

Szemerédi's regularity lemma is a useful tool in graph theory, asserting (informally) that any graph can be partitioned into a controlled number of pieces that interact with each other in a pseudorandom way. Another application of the Grothendieck inequality is to produce a partition of the vertex set that satisfies the conclusion of Szemerédi's regularity lemma, via the cut norm estimation algorithm, in time that is polynomial in the upper bound of Szemerédi's regular partition size but independent of the number of vertices in the graph.[18]

It turns out that the main "bottleneck" of constructing a Szemeredi's regular partition in polynomial time is to determine in polynomial time whether or not a given pair is close to being -regular, meaning that for all with , we have

where for all and are the vertex and edge sets of the graph, respectively. To that end, we construct an matrix , where , defined by

Then for all ,

Hence, if is not -regular, then . It follows that using the cut norm approximation algorithm together with the rounding technique, one can find in polynomial time such that

Then the algorithm for producing a Szemerédi's regular partition follows from the constructive argument of Alon et al.[20]

Variants of the Grothendieck inequality[edit]

Grothendieck inequality of a graph[edit]

The Grothendieck inequality of a graph states that for each and for each graph without self loops, there exists a universal constant such that every matrix satisfies that

[21]

The Grothendieck constant of a graph , denoted , is defined to be the smallest constant that satisfies the above property.

The Grothendieck inequality of a graph is an extension of the Grothendieck inequality because the former inequality is the special case of the latter inequality when is a bipartite graph with two copies of as its bipartition classes. Thus,

For , the -vertex complete graph, the Grothendieck inequality of becomes

It turns out that . On one hand, we have .[22][23][24] Indeed, the following inequality is true for any matrix , which implies that by the Cauchy-Schwarz inequality:[21]

On the other hand, the matching lower bound is due to Alon, Makarychev, Makarychev and Naor in 2006.[21]

The Grothendieck inequality of a graph depends upon the structure of . It is known that

[21]

and

[25]

where is the clique number of , i.e., the largest such that there exists with such that for all distinct , and

The parameter is known as the Lovász theta function of the complement of .[26][27][21]

L^p Grothendieck inequality[edit]

In the application of the Grothendieck inequality for approximating the cut norm, we have seen that the Grothendieck inequality answers the following question: How well does an optimal solution to the semidefinite program approximate , which can be viewed as an optimization problem over the unit cube? More generally, we can ask similar questions over convex bodies other than the unit cube.

For instance, the following inequality is due to Naor and Schechtman[28] and independently due to Guruswami et al:[29] For every matrix and every ,

where

The constant is sharp in the inequality. Stirling's formula implies that as .

See also[edit]

References[edit]

  1. ^ Pisier, Gilles (April 2012), "Grothendieck's Theorem, Past and Present", Bulletin of the American Mathematical Society, 49 (2): 237–323, arXiv:1101.4195, doi:10.1090/S0273-0979-2011-01348-9.
  2. ^ a b c d Grothendieck, Alexander (1953), "Résumé de la théorie métrique des produits tensoriels topologiques", Bol. Soc. Mat. Sao Paulo, 8: 1–79, MR 0094682.
  3. ^ Blei, Ron C. (1987), "An elementary proof of the Grothendieck inequality", Proceedings of the American Mathematical Society, American Mathematical Society, 100 (1): 58–60, doi:10.2307/2046119, ISSN 0002-9939, JSTOR 2046119, MR 0883401.
  4. ^ Finch, Steven R. (2003), Mathematical constants, Cambridge University Press, ISBN 978-0-521-81805-6.
  5. ^ a b c Krivine, J.-L. (1979), "Constantes de Grothendieck et fonctions de type positif sur les sphères", Advances in Mathematics, 31 (1): 16–30, doi:10.1016/0001-8708(79)90017-3, ISSN 0001-8708, MR 0521464.
  6. ^ a b Braverman, Mark; Makarychev, Konstantin; Makarychev, Yury; Naor, Assaf (2011), "The Grothendieck Constant is Strictly Smaller than Krivine's Bound", 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 453–462, arXiv:1103.6161, doi:10.1109/FOCS.2011.77.
  7. ^ Boris Tsirelson (1987). "Quantum analogues of the Bell inequalities. The case of two spatially separated domains" (PDF). Journal of Soviet Mathematics. 36 (4): 557–570. doi:10.1007/BF01663472.
  8. ^ Acín, Antonio; Gisin, Nicolas; Toner, Benjamin (2006), "Grothendieck's constant and local models for noisy entangled quantum states", Physical Review A, 73 (6): 062105, arXiv:quant-ph/0606138, Bibcode:2006PhRvA..73f2105A, doi:10.1103/PhysRevA.73.062105.
  9. ^ Davie, A. M. (1984), Unpublished.
  10. ^ Fishburn, P. C.; Reeds, J. A. (1994), "Bell Inequalities, Grothendieck's Constant, and Root Two", SIAM Journal on Discrete Mathematics, 7 (1): 48–56, doi:10.1137/S0895480191219350.
  11. ^ Vértesi, Tamás (2008), "More efficient Bell inequalities for Werner states", Physical Review A, 78 (3): 032112, arXiv:0806.0096, Bibcode:2008PhRvA..78c2112V, doi:10.1103/PhysRevA.78.032112.
  12. ^ Briët, Jop; Buhrman, Harry; Toner, Ben (2011), "A Generalized Grothendieck Inequality and Nonlocal Correlations that Require High Entanglement", Communications in Mathematical Physics, 305 (3): 827, Bibcode:2011CMaPh.305..827B, doi:10.1007/s00220-011-1280-3.
  13. ^ Hua, Bobo; Li, Ming; Zhang, Tinggui; Zhou, Chunqin; Li-Jost, Xianqing; Fei, Shao-Ming (2015), "Towards Grothendieck Constants and LHV Models in Quantum Mechanics", Journal of Physics A: Mathematical and Theoretical, Journal of Physics A, 48 (6): 065302, arXiv:1501.05507, Bibcode:2015JPhA...48f5302H, doi:10.1088/1751-8113/48/6/065302.
  14. ^ Diviánszky, Péter; Bene, Erika; Vértesi, Tamás (2017), "Qutrit witness from the Grothendieck constant of order four", Physical Review A, 96 (1): 012113, arXiv:1707.04719, Bibcode:2017PhRvA..96a2113D, doi:10.1103/PhysRevA.96.012113.
  15. ^ Rietz, Ronald E. (1974), "A proof of the Grothendieck inequality", Israel Journal of Mathematics, 19 (3): 271–276, doi:10.1007/BF02757725.
  16. ^ Hirsch, Flavien; Quintino, Marco Túlio; Vértesi, Tamás; Navascués, Miguel; Brunner, Nicolas (2017), "Better local hidden variable models for two-qubit Werner states and an upper bound on the Grothendieck constant", Quantum, 1: 3, arXiv:1609.06114, Bibcode:2016arXiv160906114H, doi:10.22331/q-2017-04-25-3.
  17. ^ Alon, Noga; Naor, Assaf (January 2006). "Approximating the Cut-Norm via Grothendieck's Inequality". SIAM Journal on Computing. 35 (4): 787–803. doi:10.1137/S0097539704441629. ISSN 0097-5397.
  18. ^ a b Khot, Subhash; Naor, Assaf (2012-04-25). "Grothendieck-Type Inequalities in Combinatorial Optimization". Communications on Pure and Applied Mathematics. 65 (7): 992–1035. doi:10.1002/cpa.21398. ISSN 0010-3640.
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  25. ^ Briet, Jop; de Oliveira Filho, Fernando Mario; Vallentin, Frank (2014). "Grothendieck Inequalities for Semidefinite Programs with Rank Constraint". Theory of Computing. 10 (1): 77–105. doi:10.4086/toc.2014.v010a004. ISSN 1557-2862.
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  27. ^ Karger, David; Motwani, Rajeev; Sudan, Madhu (1998-03-01). "Approximate graph coloring by semidefinite programming". Journal of the ACM. 45 (2): 246–265. doi:10.1145/274787.274791. ISSN 0004-5411.
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  29. ^ Guruswami, Venkatesan; Raghavendra, Prasad; Saket, Rishi; Wu, Yi (2012-01-17). "Bypassing UGC from some Optimal Geometric Inapproximability Results". Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms. Philadelphia, PA: Society for Industrial and Applied Mathematics. doi:10.1137/1.9781611973099.58.

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