Positive semidefinite rank

@article{Fawzi2014PositiveSR,
  title={Positive semidefinite rank},
  author={Hamza Fawzi and Jo{\~a}o Gouveia and Pablo A. Parrilo and Richard Z. Robinson and Rekha R. Thomas},
  journal={Mathematical Programming},
  year={2014},
  volume={153},
  pages={133-177}
}
Let $$M \in \mathbb {R}^{p \times q}$$M∈Rp×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 $$A_i, B_j$$Ai,Bj of size $$k \times k$$k×k such that $$M_{ij} = {{\mathrm{trace}}}(A_i B_j)$$Mij=trace(AiBj). 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… 

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References

SHOWING 1-10 OF 59 REFERENCES

Support-based lower bounds for the positive semidefinite rank of a nonnegative matrix

The power of lower bounds on positive semidefinite rank is characterized based on solely on the support of the matrix S, i.e., its zero/non-zero pattern.

Worst-case results for positive semidefinite rank

Using geometry and bounds on quantifier elimination, this decision can be made in polynomial time when k is fixed and it is proved that the psd rank of a generic n-dimensional polytope with v vertices is at least (nv)^{\frac{1}{4}}$$(nv)14 improving on previous lower bounds.

On the Closure of the Completely Positive Semidefinite Cone and Linear Approximations to Quantum Colorings

A hierarchy of polyhedral cones is constructed which covers the interior of the completely positive semidefinite cone $\mathcal{CS}_+^n$, which is used for computing some variants of the quantum chromatic number by way of a linear program.

A Remark on the Rank of Positive Semidefinite Matrices Subject to Affine Constraints

A short geometric proof of this result is given, which is used to improve a bound on realizability of weighted graphs as graphs of distances between points in Euclidean space, and its relation to theorems of Bohnenblust, Friedland and Loewy, and Au-Yeung and Poon.

Polytopes of Minimum Positive Semidefinite Rank

This paper shows that the psd rank of a polytope is at least the dimension of the polytopes plus one, and characterize those polytopes whose pSD rank equals this lower bound.

An Almost Optimal Algorithm for Computing Nonnegative Rank

  • Ankur Moitra
  • Computer Science, Mathematics
    SIAM J. Comput.
  • 2016
This is the first exact algorithm that runs in time singly exponential in $r$ and is built on methods for finding a solution to a system of polynomial inequalities.

Conic Approach to Quantum Graph Parameters Using Linear Optimization Over the Completely Positive Semidefinite Cone

This new cone is investigated, a new matrix cone consisting of all $n\times n$ matrices that admit a Gram representation by positive semidefinite matrices (of any size) and is used to model quantum analogues of the classical independence and chromatic graph parameters.

On the existence of 0/1 polytopes with high semidefinite extension complexity

It is shown that there is a 0/1 polytope such that any spectrahedron projecting to it must be the intersection of a semidefinite cone of dimension $$2^{\varOmega (n)}$$2Ω(n) and an affine space.
...