Positive semidefinite rank
نویسندگان
چکیده
منابع مشابه
Completely Positive Semidefinite Rank
An n×n matrix X is called completely positive semidefinite (cpsd) if there exist d×d Hermitian positive semidefinite matrices {Pi}i=1 (for some d ≥ 1) such that Xij = Tr(PiPj), for all i, j ∈ {1, . . . , n}. The cpsd-rank of a cpsd matrix is the smallest d ≥ 1 for which such a representation is possible. In this work we initiate the study of the cpsd-rank which we motivate twofold. First, the c...
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The positive semidefinite (psd) rank of a nonnegative real matrix M is the smallest integer k for which it is possible to find psd matrices Ai assigned to the rows of M and Bj assigned to the columns of M , of size k ˆ k, such that pi, jq-entry of M is the inner product of Ai and Bj . This is an example of a cone rank of a nonnegative matrix similar to nonnegative rank, and was introduced for s...
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A real symmetric matrix M is completely positive semidefinite if it admits a Gram representation by positive semidefinite matrices (of any size d). The smallest such d is called the completely positive semidefinite rank of M , and it is an open question whether there exists an upper bound on this number as a function of the matrix size. We show that if such an upper bound exists, it has to be a...
متن کاملPolytopes of Minimum Positive Semidefinite Rank
The positive semidefinite (psd) rank of a polytope is the smallest k for which the cone of 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.
متن کاملWorst-case results for positive semidefinite rank
This paper presents various worst-case results on the positive semidefinite (psd) rank of a nonnegative matrix, primarily in the context of polytopes. We prove that the psd rank of a generic n-dimensional polytope with v vertices is at least (nv) 1 4 improving on previous lower bounds. For polygons with v vertices, we show that psd rank cannot exceed 4 dv/6e which in turn shows that the psd ran...
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ژورنال
عنوان ژورنال: Mathematical Programming
سال: 2015
ISSN: 0025-5610,1436-4646
DOI: 10.1007/s10107-015-0922-1