Mathematical Programming

, Volume 153, Issue 1, pp 133–177 | Cite as

Positive semidefinite rank

  • Hamza FawziEmail author
  • João Gouveia
  • Pablo A. Parrilo
  • Richard Z. Robinson
  • Rekha R. Thomas
Full Length Paper Series B


Let \(M \in \mathbb {R}^{p \times q}\) be a nonnegative matrix. The positive semidefinite rank (psd rank) of M is the smallest integer k for which there exist positive semidefinite matrices \(A_i, B_j\) of size \(k \times k\) such that \(M_{ij} = {{\mathrm{trace}}}(A_i B_j)\). The psd rank has many appealing geometric interpretations, including semidefinite representations of polyhedra and information-theoretic applications. In this paper we develop and survey the main mathematical properties of psd rank, including its geometry, relationships with other rank notions, and computational and algorithmic aspects.

Mathematics Subject Classification

90C22 15A23 68Q17 



We thank Troy Lee for sharing his results on the psd rank of Kronecker products as well on the Hermitian psd rank. The authors also thank Thomas Rothvoß for his helpful input in Corollary 6 and his comments on an earlier draft, and Daniel Dadush for pointing out that the current stronger statement of Corollary 6 was implied by our original proof.


  1. 1.
    Arora, S., Ge, R., Kannan, R., Moitra, A.: Computing a nonnegative matrix factorization-provably. In: Proceedings of the 44th Symposium on Theory of Computing (STOC), ACM, pp. 145–162 (2012)Google Scholar
  2. 2.
    Barvinok, A.: A remark on the rank of positive semidefinite matrices subject to affine constraints. Discret. Comput. Geom. 25(1), 23–31 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Barvinok, A.: Approximations of Convex Bodies by Polytopes and by Projections of Spectrahedra. arXiv preprint. arXiv:1204.0471 (2012)
  4. 4.
    Berman, A., Shaked-Monderer, N.: Completely Positive Matrices. World Scientific Pub Co Inc, Singapore (2003)zbMATHCrossRefGoogle Scholar
  5. 5.
    Blekherman, G., Parrilo, P.A., Thomas, R. (eds.): Semidefinite Optimization and Convex Algebraic Geometry, vol 13 of MOS-SIAM Series on Optimization. SIAM (2012)Google Scholar
  6. 6.
    Bocci, C., Carlini, E., Rapallo, F.: Perturbation of matrices and nonnegative rank with a view toward statistical models. SIAM J. Matrix Anal. Appl. 32(4), 1500–1512 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Boyd, S.P., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)zbMATHCrossRefGoogle Scholar
  8. 8.
    Briët, J., Dadush, D., Pokutta, S.: On the existence of 0/1 polytopes with high semidefinite extension complexity. Algorithms. ESA 2013. Lecture Notes in Computer Science. vol 8125, pp. 217–228. Springer, Berlin (2013)Google Scholar
  9. 9.
    Brunner, N., Pironio, S., Acin, A., Gisin, N., Méthot, A.A., Scarani, V.: Testing the dimension of Hilbert spaces. Phys. Rev. Lett. 100(21), 210503 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Burgdorf, S., Laurent, M., Piovesan, T.: On the closure of the completely positive semidefinite cone and linear approximations to quantum colorings. arXiv preprint arXiv:1502.02842 (2015)
  11. 11.
    Cohen, J.E., Rothblum, U.G.: Nonnegative ranks, decompositions, and factorizations of nonnegative matrices. Linear Algebra Appl. 190, 149–168 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Dur, M.: Copositive programming-a survey. In: Diehl., M. G., Francois, J., Elias, M. W. (eds.) Recent advances in optimization and its applications in engineering, Springer, Berlin, Heidelberg, pp. 3–20 (2010). doi: 10.1007/978-3-642-12598-0_1
  13. 13.
    Fawzi, H., Saunderson, J., Parrilo, P.A.: Sparse Sum-of-Squares Certificates on Finite Abelian Groups. arXiv preprint. arXiv:1503.01207 (2015)
  14. 14.
    Fiorini, S., Massar, S., Pokutta, S., Tiwary, H.R., de Wolf, R.: Linear vs. semidefinite extended formulations: exponential separation and strong lower bounds. In: Proceedings of the Forty-fourth Annual ACM Symposium on Theory of Computing, STOC ’12, pp 95–106. ACM (2012)Google Scholar
  15. 15.
    Fiorini, S., Kaibel, V., Pashkovich, K., Theis, D.O.: Combinatorial bounds on nonnegative rank and extended formulations. Discrete Math. 313(1), 67–83 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Frenkel, P.E., Weiner, M.: On vector configurations that can berealized in the cone of positive matrices. Linear Alg. Appl. 459, 465–474 (2014). doi: 10.1016/j.laa.2014.07.017 zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A guide to the Theory of NP-Completeness. W. H Freeman and Company, San Francisco (1979)zbMATHGoogle Scholar
  18. 18.
    Gillis, N., Glineur, F.: On the geometric interpretation of the nonnegative rank. Linear Algebra Appl. 437(11), 2685–2712 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Goemans, M. X.: Smallest compact formulation for the permutahedron. Math. Prog. (2014). pp. 1–7 doi: 10.1007/s10107-014-0757-1
  20. 20.
    Gouveia, J., Parrilo, P.A., Thomas, R.R.: Lifts of convex sets and cone factorizations. Math. Operat. Res. 38(2), 248–264 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Gouveia, J., Robinson, R.Z., Thomas, R.R.: Polytopes of minimum positive semidefinite rank. Discrete Comput. Geom. 50(3), 679–699 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Gouveia, J., Fawzi, H., Robinson, R.Z.: Rational and Real Positive Semidefinite Rank can be Different. arXiv preprint. arXiv:1404.4864 (2014)
  23. 23.
    Gouveia, J., Robinson, R.Z., Thomas, R.R.:Worst-case results for positive semidefinite rank. Math. Program. 1–12 (2015). doi: 10.1007/s10107-015-0867-4
  24. 24.
    Hrubeš, P.: On the nonnegative rank of distance matrices. Inf. Process. Lett. 112, 457–461 (2012)zbMATHCrossRefGoogle Scholar
  25. 25.
    Jain, R., Shi, Y., Wei, Z., Zhang, S.: Efficient protocols for generating bipartite classical distributions and quantum states. IEEE Trans. Inf. Theory 59(8), 5171–5178 (2013)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Jolliffe, I.: Principal Component Analysis, 2nd edn. Springer, New York (2002)zbMATHGoogle Scholar
  27. 27.
    Kalman, R.E.: Mathematical description of linear dynamical systems. J. Soc. Ind. Appl. Math. Ser. A. Control 1(2), 152–192 (1963)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Laurent, M., Piovesan, T.: Conic Approach to Quantum Graph Parameters Using Linear Optimization Over the Completely Positive Semidefinite Cone. arXiv preprint. arXiv:1312.6643 (2013)
  29. 29.
    Lee, D.D., Seung, H.S.: Learning the parts of objects by non-negative matrix factorization. Nature 401(6755), 788–791 (1999)CrossRefGoogle Scholar
  30. 30.
    Lee, T., Theis, D.O.: Support-Based Lower Bounds for the Positive Semidefinite Rank of a Nonnegative Matrix. arXiv preprint. arXiv:1203.3961 (2012)
  31. 31.
    Lee, J. R., Raghavendra, P., Steurer, D.: Lower Bounds on the Size of Semidefinite Programming Relaxations. arXiv preprint. arXiv:1411.6317 (2014)
  32. 32.
    Lee, T., Wei, Z., de Wolf, R.: Some Upper and Lower Bounds on Psd-rank arXiv preprint. arXiv:1407.4308 (2014)
  33. 33.
    Linial, N., Shraibman, A.: Lower bounds in communication complexity based on factorization norms. Random Struct. Algorithms 34(3), 368–394 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Linial, N., Mendelson, S., Schechtman, G., Shraibman, A.: Complexity measures of sign matrices. Combinatorica 27(4), 439–463 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Lim, L.-H., Comon, P.: Nonnegative approximations of nonnegative tensors. J. Chemometrics 23(7–8), 432–441 (2009)Google Scholar
  36. 36.
    Moitra, A.: An Almost Optimal Algorithm for Computing Nonnegative Rank. In: Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposiumon Discrete Algorithms. chap. 104, pp. 1454–1464 (2013). doi: 10.1137/1.9781611973105.104
  37. 37.
    Mond, D., Smith, J., van Straten, D.: Stochastic factorizations, sandwiched simplices and the topology of the space of explanations. R. Soc. Proc. Math. Phys. Eng. Sci. 459(2039), 2821–2845 (2003)zbMATHCrossRefGoogle Scholar
  38. 38.
    Moore, B.: Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Autom. Control 26(1), 17–32 (1981)zbMATHCrossRefGoogle Scholar
  39. 39.
    Nie, J., Ranestad, K., Sturmfels, B.: The algebraic degree of semidefinite programming. Math. Program. 122(2), 379–405 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Pashkovich, K.: Extended formulations for combinatorial polytopes. PhD thesis, Otto-von-Guericke-Universität Magdeburg (2012)Google Scholar
  41. 41.
    Pataki, G.: On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues. Math. Operat. Res. 23(2), 339–358 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    Pólik, I., Terlaky, T.: A survey of the S-lemma. SIAM review. 49(3), 371–418 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  43. 43.
    Renegar, J.: On the computational complexity and geometry of the first-order theory of the reals. Part I: Introduction. Preliminaries. The geometry of semi-algebraic sets. The decision problem for the existential theory of the reals. J. Symb. Comput. 13(3), 255–299 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Renegar, J.: Hyperbolic programs, and their derivative relaxations. Found. Comput. Math. 6(1), 59–79 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  45. 45.
    Scheiderer, C.: Semidefinite representation for convex hulls of real algebraic curves. arXiv preprint. arXiv:1208.3865 (2012)
  46. 46.
    Stark, C. J., Harrow, A. W.: Compressibility of positive semidefinite factorizations and quantum models. arXiv preprint arXiv:1412.7437 (2014)
  47. 47.
    Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Rev. 38(1), 49–95 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  48. 48.
    Vavasis, S.A.: On the complexity of nonnegative matrix factorization. SIAM J. Optim. 20(3), 1364–1377 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    Wehner, S., Christandl, M., Doherty, A.C.: Lower bound on the dimension of a quantum system given measured data. Phys. Rev. A 78(6), 062112 (2008)CrossRefGoogle Scholar
  50. 50.
    Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. J. Comput. 43(3), 441–446 (1991)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2015

Authors and Affiliations

  • Hamza Fawzi
    • 1
    Email author
  • João Gouveia
    • 2
  • Pablo A. Parrilo
    • 1
  • Richard Z. Robinson
    • 3
  • Rekha R. Thomas
    • 3
  1. 1.Laboratory for Information and Decision Systems, Department of Electrical Engineering and Computer ScienceMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.CMUC, Department of MathematicsUniversity of CoimbraCoimbraPortugal
  3. 3.Department of MathematicsUniversity of WashingtonSeattleUSA

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