an:06494249
Zbl 1327.90174
Fawzi, Hamza; Gouveia, Jo??o; Parrilo, Pablo A.; Robinson, Richard Z.; Thomas, Rekha R.
Positive semidefinite rank
EN
Math. Program. 153, No. 1 (B), 133-177 (2015).
00348648
2015
j
90C22 15A23 68Q17
Summary: 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.