Some upper and lower bounds on PSD-rank
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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.
KeywordsSemidefinite programming Extended formulation PSD-rank Slack matrix
Mathematics Subject Classification15A23 68Q17 90C22
We would like to thank Rahul Jain for helpful discussions, and Hamza Fawzi, Richard Robinson, and Rekha Thomas for sharing their results on the derangement matrix. Troy Lee and Zhaohui Wei are supported in part by the Singapore National Research Foundation under NRF RF Award No. NRF-NRFF2013-13. Ronald de Wolf is partially supported by ERC Consolidator Grant QPROGRESS and by the EU STREP project QALGO (Grant Agreement No. 600700).
- 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
- 4.Rothvoß, T.: The matching polytope has exponential extension complexity. In: Proceedings of 46th ACM STOC, pp. 263–272 (2014)Google Scholar
- 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.Lee, T. and Wei, Z.: The square root rank of the correlation polytope is exponential (2014). arXiv:1411.6712
- 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.Zhang, S.: Quantum strategic game theory. In: Proceedings of the 3rd Innovations in Theoretical Computer Science, pp 39–59 (2012). arXiv:1012.5141
- 11.Lee, T. and Theis, D.O.: Support based bounds for positive semidefinite rank (2012). arXiv:1203.3961
- 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