Mathematical Programming

, Volume 162, Issue 1–2, pp 495–521

Some upper and lower bounds on PSD-rank

Full Length Paper Series A
  • 154 Downloads

Abstract

Positive semidefinite rank (PSD-rank) is a relatively new complexity measure on matrices, with applications to combinatorial optimization and communication complexity. We first study several basic properties of PSD-rank, and then develop new techniques for showing lower bounds on the PSD-rank. All of these bounds are based on viewing a positive semidefinite factorization of a matrix M as a quantum communication protocol. These lower bounds depend on the entries of the matrix and not only on its support (the zero/nonzero pattern), overcoming a limitation of some previous techniques. We compare these new lower bounds with known bounds, and give examples where the new ones are better. As an application we determine the PSD-rank of (approximations of) some common matrices.

Keywords

Semidefinite programming Extended formulation PSD-rank Slack matrix 

Mathematics Subject Classification

15A23 68Q17 90C22 

References

  1. 1.
    Gouveia, J., Parrilo, P., Thomas, R.: Lifts of convex sets and cone factorizations. Math. Oper. Res. 38(2), 248–264 (2013). arXiv:1111.3164 MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Fiorini, S., Massar, S., Pokutta, S., Tiwary, H. R. and de Wolf, R.: Exponential lower bounds for polytopes in combinatorial optimization. J. ACM, 16(2), 2015. Earlier version in STOC’12. arXiv:1111.0837
  3. 3.
    Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci. 43(3), 441–466 (1991)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Rothvoß, T.: The matching polytope has exponential extension complexity. In: Proceedings of 46th ACM STOC, pp. 263–272 (2014)Google Scholar
  5. 5.
    Lee, J., Steurer, D. and Raghavendra, P.: Lower bounds on the size of semidefinite programming relaxations. In: Proceedings of 47th ACM STOC, pp. 567–576 (2015). arXiv:1411.6317
  6. 6.
    Lee, T. and Wei, Z.: The square root rank of the correlation polytope is exponential (2014). arXiv:1411.6712
  7. 7.
    Conforti, M., Faenza, Y., Fiorini, S. and Tiwary, H.R.: Extended formulations, non-negative factorizations and randomized communication protocols. In: 2nd International Symposium on Combinatorial Optimization, pp. 129–140 (2012). arXiv:1105.4127
  8. 8.
    Zhang, S.: Quantum strategic game theory. In: Proceedings of the 3rd Innovations in Theoretical Computer Science, pp 39–59 (2012). arXiv:1012.5141
  9. 9.
    Jain, R., Shi, Y., Wei, Z., Zhang, S.: Efficient protocols for generating bipartite classical distributions and quantum states. IEEE Trans. Inf. Theory 59, 5171–5178 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)CrossRefMATHGoogle Scholar
  11. 11.
    Lee, T. and Theis, D.O.: Support based bounds for positive semidefinite rank (2012). arXiv:1203.3961
  12. 12.
    Sikora, J., Varvitsiotis, A. and Wei, Z.: On the minimum dimension of a Hilbert space needed to generate a quantum correlation (2015). arXiv:1507.00213
  13. 13.
    Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  14. 14.
    Ben-Tal, A., Nemirovski, A.: On polyhedral approximations of the second-order cone. Math. Oper. Res. 26(2), 193–205 (2001)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Fawzi, H., Parrilo, P.: Lower bounds on nonnegative rank via nonnegative nuclear norms. Math. Program., Ser. B 153(1), 41–66 (2015). arXiv:1210.6970 MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Fawzi, H., Gouveia, J., Parrilo, P., Robinson, R., Thomas, R.: Positive semidefinite rank. Math. Program., Ser. B 153(1), 133–177 (2015). arXiv:1407.4095 MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Alon, N.: Perturbed identity matrices have high rank: proof and applications. Combin., Prob., Comput. 18, 3–15 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016

Authors and Affiliations

  1. 1.School of Physics and Mathematical SciencesNanyang Technological University and Centre for Quantum TechnologiesSingaporeSingapore
  2. 2.QuSoftCWI and University of AmsterdamAmsterdamThe Netherlands

Personalised recommendations