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

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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

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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

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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

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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] Designolle et al., 2023 [15]
2   ≈ 1.41421
3 1.41724 1.41758 1.4359 1.43665
4 1.44521 1.44566 1.4821
5   ≈ 1.42857 1.46007 1.46112
6 1.47017
7 1.46286 1.47583
8 1.47586 1.47972
9 1.48608
10 1.49431
  ≈ 1.57079 1.67696

Upper bounds

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Some historical data on best known upper bounds of  :

d Grothendieck, 1953[2] Rietz, 1974[16] Krivine, 1979[5] Braverman et al., 2011[6] Hirsch et al., 2016[17] Designolle et al., 2023 [15]
2   ≈ 1.41421
3 1.5163 1.4644 1.4546
4   ≈ 1.5708
8 1.6641
  ≈ 2.30130 2.261   ≈ 1.78221  

Applications

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Cut norm estimation

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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.[18] 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

 [19]

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  .[20] Therefore, one can output   which satisfies

 

Szemerédi's regularity lemma

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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.[19]

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.[21]

Variants of the Grothendieck inequality

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Grothendieck inequality of a graph

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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

 [22]

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  .[23][24][25] Indeed, the following inequality is true for any   matrix  , which implies that   by the Cauchy-Schwarz inequality:[22]

 

On the other hand, the matching lower bound   is due to Alon, Makarychev, Makarychev and Naor in 2006.[22]

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

 [22]

and

 [26]

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  .[27][28][22]

L^p Grothendieck inequality

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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[29] and independently due to Guruswami et al:[30] For every   matrix   and every  ,

 

where

 

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

See also

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References

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  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, S2CID 119162963.
  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, 100 (1), American Mathematical Society: 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, S2CID 7803437.
  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. S2CID 119363229.
  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, S2CID 2588399.
  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, S2CID 119119134.
  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, 48 (6), Journal of Physics A: 065302, arXiv:1501.05507, Bibcode:2015JPhA...48f5302H, doi:10.1088/1751-8113/48/6/065302, S2CID 1082714.
  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, S2CID 119079607.
  15. ^ a b Sébastien Designolle; Gabriele Iommazzo; Mathieu Besançon; Sebastian Knebel; Patrick Gelß; Sebastian Pokutta (2023), "Improved local models and new Bell inequalities via Frank-Wolfe algorithms", Physical Review Research, 5 (4): 043059, arXiv:2302.04721, doi:10.1103/PhysRevResearch.5.043059
  16. ^ Rietz, Ronald E. (1974), "A proof of the Grothendieck inequality", Israel Journal of Mathematics, 19 (3): 271–276, doi:10.1007/BF02757725.
  17. ^ 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:2017Quant...1....3H, doi:10.22331/q-2017-04-25-3, S2CID 14199122.
  18. ^ 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.
  19. ^ 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. arXiv:1108.2464. doi:10.1002/cpa.21398. ISSN 0010-3640. S2CID 3175317.
  20. ^ P., Boyd, Stephen (2011). Convex optimization. Cambridge Univ. Pr. ISBN 978-0-521-83378-3. OCLC 767754283.{{cite book}}: CS1 maint: multiple names: authors list (link)
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  25. ^ Charikar, M.; Wirth, A. (2004). "Maximizing Quadratic Programs: Extending Grothendieck's Inequality". 45th Annual IEEE Symposium on Foundations of Computer Science. IEEE. pp. 54–60. doi:10.1109/focs.2004.39. ISBN 0-7695-2228-9. S2CID 7036076.
  26. ^ 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. arXiv:1011.1754. doi:10.4086/toc.2014.v010a004. ISSN 1557-2862. S2CID 1004947.
  27. ^ Lovasz, L. (January 1979). "On the Shannon capacity of a graph". IEEE Transactions on Information Theory. 25 (1): 1–7. doi:10.1109/TIT.1979.1055985. ISSN 0018-9448.
  28. ^ 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.
  29. ^ Naor, A., & Schechtman, G. (2009). An approximation scheme for quadratic form maximization on convex bodies. Manuscript, 1(4), 8.
  30. ^ 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: 699–717. doi:10.1137/1.9781611973099.58. ISBN 978-1-61197-210-8.
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