Matrices With High Completely Positive Semidefinite Rank

  title={Matrices With High Completely Positive Semidefinite Rank},
  author={Sander Gribling and David de Laat and Monique Laurent},
  journal={Linear Algebra and its Applications},

Correlation matrices, Clifford algebras, and completely positive semidefinite rank

ABSTRACT A symmetric matrix X is completely positive semidefinite (cpsd) if there exist positive semidefinite matrices (for some ) such that for all . The of a cpsd matrix is the smallest for which

Approximate Completely Positive Semidefinite Rank.

This paper makes use of the Approximate Carath\'eodory Theorem in order to construct an approximate matrix with a low-rank Gram representation and employs the Johnson-Lindenstrauss Lemma to improve to a logarithmic dependence of the cpsd-rank on the size.

Separability of Hermitian tensors and PSD decompositions

Hermitian tensors are natural generalizations of Hermitian matrices, while possessing rather different properties. A Hermitian tensor is separable if it has a Hermitian decomposition with only

Lower Bounds on Matrix Factorization Ranks via Noncommutative Polynomial Optimization

We use techniques from (tracial noncommutative) polynomial optimization to formulate hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In particular, we consider the

Self-dual polyhedral cones and their slack matrices

We analyze self-dual polyhedral cones and prove several properties about their slack matrices. In particular, we show that self-duality is equivalent to the existence of a positive semidefinite (PSD)

Algorithms for positive semidefinite factorization

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.

Exterior Point Method for Completely Positive Factorization

Completely positive factorization (CPF) is a critical task with applications in many fields. This paper proposes a novel method for the CPF. Based on the idea of exterior point iteration, an

Completely positive semidefinite rank

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.

Universal Rigidity of Complete Bipartite Graphs

An efficient algorithm is extended to obtain an efficient algorithm, based on a sequence of linear programs, that determines whether an input framework of a complete bipartite graph is universally rigid or not.



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 Note on Extreme Correlation Matrices

An $n\times n$ complex Hermitian or real symmetric matrix is a correlation matrix if it is positive semidefinite and all its diagonal entries equal one. The collection of all $n\times n$ correlation

Positive semidefinite rank

The main mathematical properties of psd rank are surveyed, including its geometry, relationships with other rank notions, and computational and algorithmic aspects.

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.

New results on the cp-rank and related properties of co(mpletely )positive matrices

Copositive and completely positive matrices play an increasingly important role in Applied Mathematics, namely as a key concept for approximating NP-hard optimization problems. The cone of copositive

Lifts of Convex Sets and Cone Factorizations

This paper addresses the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone and shows that the existence of a lift of a conveX set to a cone is equivalent to theexistence of a factorization of an operator associated to the set and its polar via elements in the cone and its dual.