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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

Abstract

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 

Notes

Acknowledgments

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.

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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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