# Matrices With High Completely Positive Semidefinite Rank

@article{Gribling2017MatricesWH, title={Matrices With High Completely Positive Semidefinite Rank}, author={Sander Gribling and David de Laat and Monique Laurent}, journal={Linear Algebra and its Applications}, year={2017}, volume={513}, pages={122-148} }

## 19 Citations

### Correlation matrices, Clifford algebras, and completely positive semidefinite rank

- MathematicsLinear and Multilinear Algebra
- 2018

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.

- Mathematics, Computer Science
- 2020

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

- MathematicsLinear and Multilinear Algebra
- 2021

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

- Computer Science, MathematicsFound. Comput. Math.
- 2019

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

- MathematicsArXiv
- 2022

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

- 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.

### Exterior Point Method for Completely Positive Factorization

- Mathematics
- 2021

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

- 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.

### Notes and counterexamples on positive (semi) definite properties of some matrix products

- MathematicsAin Shams Engineering Journal
- 2018

### Universal Rigidity of Complete Bipartite Graphs

- MathematicsDiscret. Comput. Geom.
- 2017

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.

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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…

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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.

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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…

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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.