Polytopes of Minimum Positive Semidefinite Rank
TLDR
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.Abstract:
The positive semidefinite (psd) rank of a polytope is the smallest $$k$$k for which the cone of $$k \times k$$k×k real symmetric psd matrices admits an affine slice that projects onto the polytope. In this paper we show that the psd rank of a polytope is at least the dimension of the polytope plus one, and we characterize those polytopes whose psd rank equals this lower bound. We give several classes of polytopes that achieve the minimum possible psd rank including a complete characterization in dimensions two and three.read more
Citations
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Lifts of Convex Sets and Cone Factorizations
TL;DR: 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.
Journal ArticleDOI
Positive semidefinite rank
TL;DR: The positive semidefinite rank (psd rank) as discussed by the authors is the smallest integer k for which there exist polyhedra of size k = 1 such that the polyhedron is polyhedrically connected with the rank of k. The psd rank has many appealing geometric interpretations, including semidefinite representations of polyhedras and information-theoretic applications.
Journal ArticleDOI
Equivariant Semidefinite Lifts and Sum-of-Squares Hierarchies
TL;DR: 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.
Posted Content
Worst-Case Results For Positive Semidefinite Rank
TL;DR: 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.
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Theta rank, levelness, and matroid minors
Francesco Grande,Raman Sanyal +1 more
TL;DR: The Theta rank and levelness, a related discrete-geometric invariant, for matroid base configurations is studied and it is shown that the class of matroids with bounded ThetaRank or levelness is closed under taking minors.
References
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