# Positive semidefinite rank

@article{Fawzi2015PositiveSR, 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={2015}, 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…

## 81 Citations

### The Complexity of Positive Semidefinite Matrix Factorization

- Mathematics, Computer ScienceSIAM J. Optim.
- 2017

This paper determines the computational complexity status of the PSD rank and shows that the problem of computing this function is polynomial-time equivalent to the existential theory of the reals.

### Completely positive semidefinite rank

- Mathematics, Computer ScienceMath. Program.
- 2018

The cpsd-rank is a natural non-commutative analogue of the completely positive rank of a completely positive matrix and every doubly nonnegative matrix whose support is given by G is cPSd, and it is shown that a graph is cpsD if and only if it has no odd cycle of length at least 5 as a subgraph.

### Algorithms for positive semidefinite factorization

- Computer Science, MathematicsComput. Optim. Appl.
- 2018

This work introduces several local optimization schemes to tackle the problem of positive semidefinite factorization, a generalization of exact nonnegative matrix factorization and introduces a fast projected gradient method and two algorithms based on the coordinate descent framework.

### Optimal Size of Linear Matrix Inequalities in Semidefinite Approaches to Polynomial Optimization

- MathematicsSIAM J. Appl. Algebra Geom.
- 2019

It is shown that the cone of $k \times k$ symmetric positive semidefinite matrices has no extended formulation with finitely many LMIs of size less than $k$ and the standard extended formulation of $\Sigma_{n,2d}$ is optimal in terms of the size of the LMIs.

### Two Results on the Size of Spectrahedral Descriptions

- MathematicsSIAM J. Optim.
- 2016

It is shown that if $A,B,C \in {Sym}_r(\mathbb{R})$ are real symmetric matrices such that $f(x,y,z)=\det(I_r+A x+B y+C z)$ is a cubic polynomial, then the surface in complex projective three-space with affine equation $f-r=0$ is singular.

### A lower bound on the positive semidefinite rank of convex bodies

- Mathematics, Computer ScienceSIAM J. Appl. Algebra Geom.
- 2018

It is shown that the positive semidefinite rank of any convex body $C$ is at least $\sqrt{\log d}$, where $d$ is the smallest degree of a polynomial that vanishes on the boundary of the polar of $C$.

### A universality theorem for nonnegative matrix factorizations

- Mathematics, Computer Science
- 2016

It is shown that every bounded semialgebraic set $U$ is rationally equivalent to the set of nonnegative size-$k$ factorizations of some matrix $A$ up to a permutation of matrices in the factorization.

### Sparse sums of squares on finite abelian groups and improved semidefinite lifts

- MathematicsMath. Program.
- 2016

It is proved that any nonnegative quadratic form in n binary variables is a sum of squares of functions of degree at most, establishing a conjecture of Laurent.

### Equivariant Semidefinite Lifts and Sum-of-Squares Hierarchies

- MathematicsSIAM J. Optim.
- 2015

A representation-theoretic framework is presented to study equivariant PSD lifts of a certain class of symmetric polytopes known as orbitopes which respect the symmetries of the polytope.

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