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Completely positive semidefinite rank

Anupam Prakash, Jamie Sikora, Antonios Varvitsiotis, Zhaohui Wei
2017 Mathematical programming  
First, the cpsd-rank is a natural non-commutative analogue of the completely positive rank of a completely positive matrix.  ...  An n× n matrix X is called completely positive semidefinite (cpsd) if there exist d× d Hermitian positive semidefinite matrices {P_i}_i=1^n (for some d> 1) such that X_ij= Tr(P_iP_j), for all i,j ∈{ 1,  ...  semidefinite cone (resp. completely positive cone).  ... 
doi:10.1007/s10107-017-1198-4 fatcat:zh7h7tje2nffdkgrbpbi2eojei

Approximate Completely Positive Semidefinite Rank [article]

Paria Abbasi, Andreas Klingler, Tim Netzer
2020 arXiv   pre-print
In this paper we provide an approximation for completely positive semidefinite (cpsd) matrices with cpsd-rank bounded above (almost) independently from the cpsd-rank of the initial matrix.  ...  We then employ the Johnson-Lindenstrauss Lemma to improve to a logarithmic dependence of the cpsd-rank on the size.  ...  Our main result yields an approximation of completely positive semidefinite matrices of relatively small completely positive semidefinite rank.  ... 
arXiv:2012.06471v2 fatcat:ek46dxel2bhc7iwqqn7ommtc44

Matrices with high completely positive semidefinite rank

Sander Gribling, David de Laat, Monique Laurent
2017 Linear Algebra and its Applications  
We construct completely positive semidefinite matrices of size 4k^2+2k+2 with complex completely positive semidefinite rank 2^k for any positive integer k.  ...  We also exhibit a class of completely positive matrices with quadratic (in terms of the matrix size) completely positive rank, but with linear completely positive semidefinite rank, and we make a connection  ...  The completely positive and completely positive semidefinite ranks are symmetric analogs of the nonnegative and positive semidefinite ranks.  ... 
doi:10.1016/j.laa.2016.10.015 fatcat:h6mf372z6needfeaznss2ffcqm

Correlation matrices, Clifford algebras, and completely positive semidefinite rank [article]

Anupam Prakash, Antonios Varvitsiotis
2018 arXiv   pre-print
Using this we give a self-contained and succinct proof of the existence of completely positive semidefinite matrices with sub-exponential cpsd-rank, recently derived in the literature.  ...  We show that for extremal correlations, the matrices in such a factorization generate a Clifford algebra and thus, their size is exponential in terms of the rank of the correlation matrix.  ...  The completely positive semidefinite rank of X ∈ CS n + , denoted by cpsd-rank(X), is defined as the least d ≥ 1 for which there exist d × d Hermitian positive semidefinite matrices P 1 , . . . , P n satisfying  ... 
arXiv:1702.06305v2 fatcat:g3qqobyupvehxeynelhhaxa7la

Complexity of the positive semidefinite matrix completion problem with a rank constraint [article]

Marianna Eisenberg-Nagy, Monique Laurent, Antonios Varvitsiotis
2012 arXiv   pre-print
completed to a positive semidefinite matrix of rank at most k.  ...  We consider the decision problem asking whether a partial rational symmetric matrix with an all-ones diagonal can be completed to a full positive semidefinite matrix of rank at most k.  ...  entries are all equal to 1, that can be completed to a positive semidefinite matrix.  ... 
arXiv:1203.6602v2 fatcat:hi5mdc72nngyzauui5v73rlpxa

A new graph parameter related to bounded rank positive semidefinite matrix completions [article]

Monique Laurent, Antonios Varvitsiotis
2012 arXiv   pre-print
to edges of G, can be completed to a positive semidefinite matrix of rank at most k (assuming a positive semidefinite completion exists).  ...  The Gram dimension (G) of a graph G is the smallest integer k> 1 such that any partial real symmetric matrix, whose entries are specified on the diagonal and at the off-diagonal positions corresponding  ...  The problem of completing a partial matrix to a full positive semidefinite (psd) matrix is one of the most extensively studied matrix completion problems.  ... 
arXiv:1204.0734v1 fatcat:y7eue5ijkrejzg2j7dz5uy7aue

A new graph parameter related to bounded rank positive semidefinite matrix completions

Monique Laurent, Antonios Varvitsiotis
2013 Mathematical programming  
to edges of G, can be completed to a positive semidefinite matrix of rank at most k (assuming a positive semidefinite completion exists).  ...  The Gram dimension gd(G) of a graph G is the smallest integer k ≥ 1 such that any partial real symmetric matrix, whose entries are specified on the diagonal and at the off-diagonal positions corresponding  ...  The problem of completing a partial matrix to a full positive semidefinite (psd) matrix is one of the most extensively studied matrix completion problems.  ... 
doi:10.1007/s10107-013-0648-x fatcat:fcsomo7egnh6dpqmm76x2slqmi

Complexity of the Positive Semidefinite Matrix Completion Problem with a Rank Constraint [chapter]

Marianna E.-Nagy, Monique Laurent, Antonios Varvitsiotis
2013 Fields Institute Communications  
can be completed to a positive semidefinite matrix of rank at most k.  ...  We consider the decision problem asking whether a partial rational symmetric matrix with an all-ones diagonal can be completed to a full positive semidefinite matrix of rank at most k.  ...  entries are all equal to 1, that can be completed to a positive semidefinite matrix.  ... 
doi:10.1007/978-3-319-00200-2_7 fatcat:5mpvfe5zkfhahjyfii7ut3jw2a

A Riemannian geometry with complete geodesics for the set of positive semidefinite matrices of fixed rank

B. Vandereycken, P.- A. Absil, S. Vandewalle
2012 IMA Journal of Numerical Analysis  
We present a homogeneous space geometry for the manifold of symmetric positive semidefinite matrices of fixed rank.  ...  Abstract We present a homogeneous space geometry for the manifold of symmetric positive semidefinite matrices of fixed rank.  ...  Sepulchre for helpful discussions about the geometry of fixed-rank matrices.  ... 
doi:10.1093/imanum/drs006 fatcat:jeu2kstwcfcivl4u27twivg3ky

Positive semidefinite completions of partial Hermitian matrices

Jerome Dancis
1992 Linear Algebra and its Applications  
We classify the ranks of positive semidefinite completions of Hermitian band matrices and other partially specified Hermitian matrices with chordal graphs and specified main diagonals.  ...  Completing a partially specified matrix means filling in the unspecified entries.  ...  there are exactly 2" real symmetric positive semidefinite completions F of R such that Rank F = Rank R.  ... 
doi:10.1016/0024-3795(92)90304-s fatcat:p4r6iihjnff5tgewi2ngpie2eu

Matrix completion and semidefinite matrices [article]

Olaf Dreyer
2021 arXiv   pre-print
Positive semidefinite Hermitian matrices that are not fully specified can be completed provided their underlying graph is chordal.  ...  For a class of matrices that are singular but of maximal rank unique characterizations can be given, just as in the positive definite case.  ...  We have shown in the last theorem that H is positive semidefinite if we set X = BC + D. H is also of maximal rank. In general we have rank H ≤ rank A + rank C + rank E.  ... 
arXiv:2112.03758v1 fatcat:zosk467cyjamdbtc4shrcx4mte

A New Algorithm for Positive Semidefinite Matrix Completion

Fangfang Xu, Peng Pan
2016 Journal of Applied Mathematics  
Positive semidefinite matrix completion (PSDMC) aims to recover positive semidefinite and low-rank matrices from a subset of entries of a matrix.  ...  This task can be conducted by solving the nuclear norm regularized linear least squares model with positive semidefinite constraints.  ...  Acknowledgments The authors would like to thank Professor Zaiwen Wen for the discussions on matrix completion. They thank Professor Xiaoming Yuan for offering the original codes of ADMM-VI.  ... 
doi:10.1155/2016/1659019 fatcat:az2re62rajb3zm7embqthof4tu

Positive semidefinite maximum nullity and zero forcing number

Travis Peters
2012 The Electronic Journal of Linear Algebra  
In addition, this technique was used in [3] to establish the minimum rank of the complete ciclo C t (K r ), which is in fact the minimum positive semidefinite rank.  ...  The minimum positive semidefinite rank of G is the smallest possible rank over all real positive semidefinite matrices described by G and the minimum Hermitian positive semidefinite rank of G is the smallest  ... 
doi:10.13001/1081-3810.1559 fatcat:yxecfsv6ube2lmnhzddsadt4lq

Semidefiniteness without real symmetry

Charles R. Johnson, Robert Reams
2000 Linear Algebra and its Applications  
part, and A 2 has positive semidefinite symmetric part, then rank[AX] = rank[X T AX] for all X ∈ M n (R).  ...  . , A k (k 2) all have positive semidefinite symmetric part, then rank[AX] = rank[X T AX] = · · · = rank[X T A k−1 X] for all X ∈ M n (R).  ...  Finally, as in the proof of Theorem 4, rank[X T A k−1 X] = rank[AX], and the induction step is complete.  ... 
doi:10.1016/s0024-3795(99)00256-6 fatcat:bdlmcw3vybcmhmsowq4rxezt6q

A direct proof for the matrix decomposition of chordal-structured positive semidefinite matrices

Naonori Kakimura
2010 Linear Algebra and its Applications  
This decomposition is recently exploited to solve positive semidefinite programming efficiently.  ...  Agler, Helton, McCullough, and Rodman proved that a graph is chordal if and only if any positive semidefinite (PSD) symmetric matrix, whose nonzero entries are specified by a given graph, can be decomposed  ...  The matrix L is nonsingular and A is positive semidefinite, which implies that X and A are both positive semidefinite.  ... 
doi:10.1016/j.laa.2010.04.012 fatcat:jkn36a2c4zaalh74wfho4l73pq
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