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arXiv · 1407.4095

Positive semidefinite rank

Hamza Fawzi·João Gouveia·Pablo A. Parrilo·Richard Z. Robinson·Rekha R. Thomas
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Abstract

Let M be a p-by-q matrix with nonnegative entries. 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 $M_{ij} = \text{trace}(A_i B_j)$. 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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Hamza Fawzi, João Gouveia, Pablo A. Parrilo, Richard Z. Robinson, Rekha R. Thomas. 2014-07-15. Positive semidefinite rank. https://doi.org/10.1007/s10107-015-0922-1

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