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Positive semidefinite rank

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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.

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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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Authors and Affiliations

  1. Laboratory for Information and Decision Systems, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Hamza Fawzi & Pablo A. Parrilo

  2. CMUC, Department of Mathematics, University of Coimbra, 3001-454, Coimbra, Portugal

    João Gouveia

  3. Department of Mathematics, University of Washington, Box 354350, Seattle, WA, 98195, USA

    Richard Z. Robinson & Rekha R. Thomas

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  1. Hamza Fawzi
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  2. João Gouveia
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Correspondence to Hamza Fawzi.

Additional information

Fawzi and Parrilo were supported in part by AFOSR FA9550-11-1-0305. Gouveia was supported by the Centre for Mathematics at the University of Coimbra and Fundação para a Ciência e a Tecnologia, through the European program COMPETE/FEDER. Robinson was supported by the US NSF Graduate Research Fellowship through Grant DGE-1256082 and Thomas was supported by NSF Grant DMS-1115293.

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Fawzi, H., Gouveia, J., Parrilo, P.A. et al. Positive semidefinite rank. Math. Program. 153, 133–177 (2015). https://doi.org/10.1007/s10107-015-0922-1

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