# 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…

## 91 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 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.

### A spectrahedral representation of the first derivative relaxation of the positive semidefinite cone

- MathematicsOptim. Lett.
- 2018

The construction provides a new explicit example of a hyperbolicity cone that is also a spectrahedron, consistent with the generalized Lax conjecture.

## References

SHOWING 1-10 OF 59 REFERENCES

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

- Computer Science, Mathematics
- 2012

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

- Mathematics, Computer ScienceMath. Program.
- 2015

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

- MathematicsTQC
- 2015

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

- MathematicsDiscret. Comput. Geom.
- 2001

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

- MathematicsDiscret. Comput. Geom.
- 2013

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

- Computer Science, MathematicsSIAM 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

- MathematicsSIAM J. Optim.
- 2015

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

- MathematicsMath. Program.
- 2013

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.