Computing Approximate PSD Factorizations

Authors Amitabh Basu, Michael Dinitz, Xin Li

Thumbnail PDF


  • Filesize: 488 kB
  • 12 pages

Document Identifiers

Author Details

Amitabh Basu
Michael Dinitz
Xin Li

Cite AsGet BibTex

Amitabh Basu, Michael Dinitz, and Xin Li. Computing Approximate PSD Factorizations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 2:1-2:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


We give an algorithm for computing approximate PSD factorizations of nonnegative matrices. The running time of the algorithm is polynomial in the dimensions of the input matrix, but exponential in the PSD rank and the approximation error. The main ingredient is an exact factorization algorithm when the rows and columns of the factors are constrained to lie in a general polyhedron. This strictly generalizes nonnegative matrix factorizations which can be captured by letting this polyhedron to be the nonnegative orthant.
  • PSD rank
  • PSD factorizations


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Sanjeev Arora, Rong Ge, Ravindran Kannan, and Ankur Moitra. Computing a nonnegative matrix factorization-provably. In Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of computing, STOC'12, pages 145-162. ACM, 2012. Google Scholar
  2. Saugata Basu, Richard Pollack, and Marie-Françoise Roy. On the combinatorial and algebraic complexity of quantifier elimination. Journal of the ACM (JACM), 43(6):1002-1045, 1996. Google Scholar
  3. Michael W Berry, Murray Browne, Amy N Langville, V Paul Pauca, and Robert J Plemmons. Algorithms and applications for approximate nonnegative matrix factorization. Computational statistics &data analysis, 52(1):155-173, 2007. Google Scholar
  4. Lenore Blum, Mike Shub, and Steve Smale. On a theory of computation and complexity over the real numbers: W-completeness, recursive functions and universal machines. Bull. Amer. Math. Soc, 21(1):1-46, 1989. Google Scholar
  5. Gábor Braun, Jonah Brown-Cohen, Arefin Huq, Sebastian Pokutta, Prasad Raghavendra, Aurko Roy, Benjamin Weitz, and Daniel Zink. The matching problem has no small symmetric sdp. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'16, pages 1067-1078, Philadelphia, PA, USA, 2016. Society for Industrial and Applied Mathematics. URL:
  6. Jop Briët, Daniel Dadush, and Sebastian Pokutta. On the existence of 0/1 polytopes with high semidefinite extension complexity. Mathematical Programming, pages 1-21, 2014. Google Scholar
  7. Michele Conforti, Gérard Cornuéjols, and Giacomo Zambelli. Extended formulations in combinatorial optimization. 4OR, 8(1):1-48, 2010. Google Scholar
  8. Hamza Fawzi, Jo ao Gouveia, Pablo A. Parrilo, Richard Z. Robinson, and Rekha R. Thomas. Positive semidefinite rank., 2015. Google Scholar
  9. Samuel Fiorini, Volker Kaibel, Kanstantsin Pashkovich, and Dirk Oliver Theis. Combinatorial bounds on nonnegative rank and extended formulations. Discrete mathematics, 313(1):67-83, 2013. Google Scholar
  10. Samuel Fiorini, Serge Massar, Sebastian Pokutta, Hans Raj Tiwary, and Ronald de Wolf. Linear vs. semidefinite extended formulations: exponential separation and strong lower bounds. In Proceedings of the 44th Annual ACM Symposium on Theory of Computing, pages 95-106. ACM, 2012. Google Scholar
  11. Joao Gouveia, Pablo A Parrilo, and Rekha R Thomas. Lifts of convex sets and cone factorizations. Mathematics of Operations Research, 38(2):248-264, 2013. Google Scholar
  12. João Gouveia, Pablo A Parrilo, and Rekha R Thomas. Approximate cone factorizations and lifts of polytopes. Mathematical Programming, 151(2):613-637, 2015. Google Scholar
  13. D Yu Grigor'ev and NN Vorobjov. Solving systems of polynomial inequalities in subexponential time. Journal of symbolic computation, 5(1):37-64, 1988. Google Scholar
  14. Didier Henrion and Jérôme Malick. Projection methods in conic optimization. In Handbook on Semidefinite, Conic and Polynomial Optimization, pages 565-600. Springer, 2012. Google Scholar
  15. Volker Kaibel. Extended formulations in combinatorial optimization. arXiv preprint arXiv:1104.1023, 2011. Google Scholar
  16. J Lee, Prasad Raghavendra, David Steurer, and Ning Tan. On the power of symmetric lp and sdp relaxations. In Proceedings of the 29th Conference on Computational Complexity (CCC), pages 13-21. IEEE, 2014. Google Scholar
  17. James R. Lee, Prasad Raghavendra, and David Steurer. Lower bounds on the size of semidefinite programming relaxations. In Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing, STOC'15, pages 567-576, New York, NY, USA, 2015. ACM. URL:
  18. Ankur Moitra. An almost optimal algorithm for computing nonnegative rank. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'13, pages 1454-1464. SIAM, 2013. Google Scholar
  19. Victor Y Pan and Zhao Q Chen. The complexity of the matrix eigenproblem. In Proceedings of the Thirty-First Annual ACM Symposium on Theory of computing, pages 507-516. ACM, 1999. Google Scholar
  20. Yuval Rabani and Amir Shpilka. Explicit construction of a small ε-net for linear threshold functions. SIAM Journal on Computing, 39(8):3501-3520, 2010. Google Scholar
  21. Thomas Rothvoß. Some 0/1 polytopes need exponential size extended formulations. Mathematical Programming, 142(1-2):255-268, 2013. Google Scholar
  22. Thomas Rothvoß. The matching polytope has exponential extension complexity. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pages 263-272. ACM, 2014. Google Scholar
  23. Stephen A Vavasis. On the complexity of nonnegative matrix factorization. SIAM Journal on Optimization, 20(3):1364-1377, 2009. Google Scholar
  24. Mihalis Yannakakis. Expressing combinatorial optimization problems by linear programs. In Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, pages 223-228. ACM, 1988. Google Scholar
  25. G. M. Ziegler. Lectures on Polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail