Grothendieck Inequalities for Semidefinite Programs with Rank Constraint

Grothendieck inequalities are fundamental inequalities which are frequently used in many areas of mathematics and computer science. They can be interpreted as upper bounds for the integrality gap between two optimization problems: a difficult semidefinite program with rank-1 constraint and its easy semidefinite relaxation where the rank constraint is dropped. For instance, the integrality gap of the Goemans-Williamson approximation algorithm for MAX CUT can be seen as a Grothendieck inequality. In this paper we consider Grothendieck inequalities for ranks greater than 1 and we give two applications: approximating ground states in the n-vector model in statistical mechanics and XOR games in quantum information theory.

[1]  H. Buhrman,et al.  A generalized Grothendieck inequality and entanglement in XOR games , 2009, 0901.2009.

[2]  J. H. PEARCE,et al.  Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen , 1945, Nature.

[3]  Subhash Khot,et al.  Approximate Kernel Clustering , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[4]  Moses Charikar,et al.  Maximizing quadratic programs: extending Grothendieck's inequality , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[5]  P. Grangier,et al.  Experimental Tests of Realistic Local Theories via Bell's Theorem , 1981 .

[6]  Noga Alon,et al.  Quadratic forms on graphs , 2005, STOC '05.

[7]  R. Muirhead.  Aspects of Multivariate Statistical Theory , 1982, Wiley Series in Probability and Statistics.

[8]  Prasad Raghavendra,et al.  Towards computing the Grothendieck constant , 2009, SODA.

[9]  J. Diestel,et al.  Absolutely Summing Operators , 1995 .

[10]  Noga Alon,et al.  Approximating the cut-norm via Grothendieck's inequality , 2004, STOC '04.

[11]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[12]  J. Krivine.  Constantes de Grothendieck et fonctions de type positif sur les sphères , 1979 .

[13]  Arkadi Nemirovski,et al.  On Tractable Approximations of Uncertain Linear Matrix Inequalities Affected by Interval Uncertainty , 2002, SIAM J. Optim..

[14]  Y. Nesterov.  Semidefinite relaxation and nonconvex quadratic optimization , 1998 .

[15]  P. Morse,et al.  Methods of theoretical physics , 1955 .

[16]  J. Swart,et al.  The Metric Theory of Tensor Products (Grothendieck's Résumé Revisited) Part 1: Tensor Norms , 2002 .

[17]  Leon M. Hall,et al.  Special Functions , 1998 .

[18]  Assaf Naor,et al.  Krivine schemes are optimal , 2012, 1205.6415.

[19]  G. Jameson.  Summing and nuclear norms in Banach space theory , 1987 .

[20]  L. Debnath.  Tables of Integral Transforms , 2012 .

[21]  D. Garling,et al.  Inequalities: A Journey into Linear Analysis , 2007 .

[22]  Mark Braverman,et al.  The Grothendieck Constant is Strictly Smaller than Krivine's Bound , 2011, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[23]  Frank Vallentin,et al.  The Positive Semidefinite Grothendieck Problem with Rank Constraint , 2009, ICALP.

[24]  Tamás Terlaky,et al.  On maximization of quadratic form over intersection of ellipsoids with common center , 1999, Math. Program..

[25]  Peter Høyer,et al.  Consequences and limits of nonlocal strategies , 2004, Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004..

[26]  A. Grothendieck.  Résumé de la théorie métrique des produits tensoriels topologiques , 1996 .

[27]  Guy Kindler,et al.  On non-approximability for quadratic programs , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[28]  Subhash Khot,et al.  Sharp kernel clustering algorithms and their associated Grothendieck inequalities , 2009, SODA '10.

[29]  J. Lindenstrauss,et al.  Absolutely summing operators in Lp spaces and their applications , 1968 .

[30]  Oded Regev,et al.  Simulating Quantum Correlations with Finite Communication , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[31]  Rolf Dach,et al.  Technical Report 2012 , 2013 .

[32]  Subhash Khot,et al.  SDP gaps and UGC-hardness for MAXCUTGAIN , 2006, IEEE Annual Symposium on Foundations of Computer Science.

[33]  Frank Vallentin,et al.  The Grothendieck problem with rank constraint , 2010 .

[34]  J. Bell.  On the Einstein-Podolsky-Rosen paradox , 1964 .

[35]  Frank E. Grubbs,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[36]  B. Tsirelson.  Quantum analogues of the Bell inequalities. The case of two spatially separated domains , 1987 .

[37]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[38]  Adi Shraibman,et al.  Lower bounds in communication complexity based on factorization norms , 2009 .

[39]  Guy Kindler,et al.  The UGC hardness threshold of the ℓp Grothendieck problem , 2008, SODA '08.

[40]  U. Haagerup.  A new upper bound for the complex Grothendieck constant , 1987 .

[41]  A. Erdélyi,et al.  Tables of integral transforms , 1955 .

[42]  Stephen P. Boyd,et al.  Semidefinite Programming , 1996, SIAM Rev..

[43]  G. Pisier.  Factorization of Linear Operators and Geometry of Banach Spaces , 1986 .

[44]  Hermann König,et al.  On an Extremal Problem Originating in Questions of Unconditional Convergence , 2001 .

[45]  Joe Diestel,et al.  The metric theory of tensor products , 2008 .

[46]  I. J. Schoenberg.  Positive definite functions on spheres , 1942 .

[47]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[48]  V. Scarani,et al.  Testing the dimension of Hilbert spaces. , 2008, Physical review letters.

[49]  D. Owen.  Handbook of Mathematical Functions with Formulas , 1965 .

[50]  Joe Diestel,et al.  The Metric Theory of Tensor Products (Grothendieck's Résumé Revisited) Part 3: Vector Sequence Spaces , 2002 .

[51]  H. Buhrman,et al.  A Generalized Grothendieck Inequality and Nonlocal Correlations that Require High Entanglement , 2011 .

[52]  A. EINsTEIN,et al.  Can Quantum-Mechanical Description of Physical Reality Be Considered Complete ' ? , 2011 .

[53]  KindlerGuy,et al.  The UGC Hardness Threshold of the Lp Grothendieck Problem , 2010 .

[54]  Uri Zwick,et al.  Rounding Two and Three Dimensional Solutions of the SDP Relaxation of MAX CUT , 2005, APPROX-RANDOM.

[55]  László Lovász,et al.  On the Shannon capacity of a graph , 1979, IEEE Trans. Inf. Theory.

[56]  Joe Diestel,et al.  The Metric Theory of Tensor Products (Grothendieck's Résumé Revisited) Part 5: Injective and Projective Tensor Norms , 2003 .

[57]  N. Alon,et al.  Quadratic forms on graphs , 2006 .

[58]  Subhash Khot,et al.  Linear Equations Modulo 2 and the L1 Diameter of Convex Bodies , 2008, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[59]  A. Shimony,et al.  Proposed Experiment to Test Local Hidden Variable Theories. , 1969 .

[60]  A. Megretski.  Relaxations of Quadratic Programs in Operator Theory and System Analysis , 2001 .

[61]  H. E. Stanley,et al.  Spherical Model as the Limit of Infinite Spin Dimensionality , 1968 .

[62]  Noga Alon,et al.  Approximating the Cut-Norm via Grothendieck's Inequality , 2006, SIAM J. Comput..