The Minimum Rank of a Correlation Matrix

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The complexity of computing the minimum rank of a sign pattern matrix

We show that computing the minimum rank of a sign pattern matrix is NP hard. Our proof is based on a simple but useful connection between minimum ranks of sign pattern matrices and the stretchability problem for pseudolines arrangements. In fact, our hardness result shows that it is already hard to determine if the minimum rank of a sign pattern matrix is ď 3. We complement this by giving a pol...

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Ela a Note on the Positive Semidefinite Minimum Rank of a Sign Pattern Matrix

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Approximation of rank function and its application to the nearest low-rank correlation matrix

The rank function rank(·) is neither continuous nor convex which brings much difficulty to the solution of rank minimization problems. In this paper, we provide a unified framework to construct the approximation functions of rank(·), and study their favorable properties. Particularly, with two families of approximation functions, we propose a convex relaxation method for the rank minimization p...

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A sign pattern matrix is a matrix whose entries are from the set {+,−, 0}. The minimum rank of a sign pattern matrix A is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of A. It is conjectured that the minimum rank of every sign pattern matrix can be realized by a rational matrix. The equivalence of this conjecture to several seemingly ...

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ژورنال

عنوان ژورنال: Proceedings of the National Academy of Sciences

سال: 1944

ISSN: 0027-8424,1091-6490

DOI: 10.1073/pnas.30.6.144