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The Procrustes Problem for Orthogonal Stiefel Matrices
 SIAM J. Scientific Computing
, 1998
"... In this paper we consider the Procrustes problem on the manifold of orthogonal Stiefel matrices. Given matrices A 2 R m\Thetak , B 2 R m\Thetap , m p k, we seek the minimum of kA \Gamma BQk 2 for all matrices Q 2 R p\Thetak , Q T Q = I k\Thetak . We introduce a class of relaxation methods for genera ..."
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Cited by 6 (0 self)
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In this paper we consider the Procrustes problem on the manifold of orthogonal Stiefel matrices. Given matrices A 2 R m\Thetak , B 2 R m\Thetap , m p k, we seek the minimum of kA \Gamma BQk 2 for all matrices Q 2 R p\Thetak , Q T Q = I k\Thetak . We introduce a class of relaxation methods
Correlation between orthogonally projected matrices
"... Abstract — We consider the problem of finding the optimal correlation between two projected matrices U ∗ AU and V ∗ BV. The square matrices A and B may be of different dimensions, but the isometries U and V have a common column dimension k. The correlation is measured by the real function c(U,V) = ℜ ..."
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Abstract — We consider the problem of finding the optimal correlation between two projected matrices U ∗ AU and V ∗ BV. The square matrices A and B may be of different dimensions, but the isometries U and V have a common column dimension k. The correlation is measured by the real function c
A NOTE ON INVERSEORTHOGONAL TOEPLITZ MATRICES
, 2013
"... In this note, inverseorthogonal Toeplitz matrices are investigated, and it is proved that every such a matrix is equivalent to a circulant one. As a corollary, it is showed that a real Hadamard matrix of order n> 2 with Toeplitz structure is necessarily circulant. ..."
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In this note, inverseorthogonal Toeplitz matrices are investigated, and it is proved that every such a matrix is equivalent to a circulant one. As a corollary, it is showed that a real Hadamard matrix of order n> 2 with Toeplitz structure is necessarily circulant.
Schur flows for orthogonal Hessenberg matrices. Hamiltonian and gradient flows, algorithms and control
 Algorithms and Control, Fields Inst. Commun
, 1994
"... Abstract We consider a standard matrix flow on the set of unitary upper Hessenberg matrices with nonnegative subdiagonal elements. The Schur parametrization of this set of matrices leads to ordinary differential equations for the weights and the parameters that are analogous with the Toda flow as id ..."
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Cited by 16 (0 self)
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as identified with a flow on Jacobi matrices. We derive explicit differential equations for the flow on the Schur parameters of orthogonal Hessenberg matrices. We also outline an efficient procedure for computing the solution of Jacobi flows and Schur flows.
MeanField Equations for Spin Models with Orthogonal Interaction Matrices
 Matrices, J. Phys. A (Math. Gen
, 1995
"... We study the metastable states in Ising spin models with orthogonal interaction matrices. We focus on three realizations of this model, the random case and two nonrandom cases, i.e. the fullyfrustrated model on an infinite dimensional hypercube and the socalled sinemodel. We use the meanfield ( ..."
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Cited by 18 (4 self)
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We study the metastable states in Ising spin models with orthogonal interaction matrices. We focus on three realizations of this model, the random case and two nonrandom cases, i.e. the fullyfrustrated model on an infinite dimensional hypercube and the socalled sinemodel. We use the mean
Finding community structure in networks using the eigenvectors of matrices
, 2006
"... We consider the problem of detecting communities or modules in networks, groups of vertices with a higherthanaverage density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as “modularity ” over possible div ..."
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Cited by 500 (0 self)
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We consider the problem of detecting communities or modules in networks, groups of vertices with a higherthanaverage density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as “modularity ” over possible divisions of a network. Here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix, which plays a role in community detection similar to that played by the graph Laplacian in graph partitioning calculations. This result leads us to a number of possible algorithms for detecting community structure, as well as several other results, including a spectral measure of bipartite structure in networks and a new centrality measure that identifies those vertices that occupy central positions within the communities to which they belong. The algorithms and measures proposed are illustrated with applications to a variety of realworld complex networks.
LINEAR ALGEBRA AND ITS APPLICATIONS www.elsevier.comllocate/laa Orthogonality of matrices
, 2001
"... Let A and B be rectangular matrices. Then A is orthogonal to B if II A + fLB II?: II A II for every scalar fL. Some approximation theory and convexity results on matrices are used to study orthogonality ..."
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Let A and B be rectangular matrices. Then A is orthogonal to B if II A + fLB II?: II A II for every scalar fL. Some approximation theory and convexity results on matrices are used to study orthogonality
GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems
 SIAM J. SCI. STAT. COMPUT
, 1986
"... We present an iterative method for solving linear systems, which has the property ofminimizing at every step the norm of the residual vector over a Krylov subspace. The algorithm is derived from the Arnoldi process for constructing an l2orthogonal basis of Krylov subspaces. It can be considered a ..."
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Cited by 2046 (40 self)
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We present an iterative method for solving linear systems, which has the property ofminimizing at every step the norm of the residual vector over a Krylov subspace. The algorithm is derived from the Arnoldi process for constructing an l2orthogonal basis of Krylov subspaces. It can be considered
On Spectral Clustering: Analysis and an algorithm
 ADVANCES IN NEURAL INFORMATION PROCESSING SYSTEMS
, 2001
"... Despite many empirical successes of spectral clustering methods  algorithms that cluster points using eigenvectors of matrices derived from the distances between the points  there are several unresolved issues. First, there is a wide variety of algorithms that use the eigenvectors in slightly ..."
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Cited by 1697 (13 self)
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Despite many empirical successes of spectral clustering methods  algorithms that cluster points using eigenvectors of matrices derived from the distances between the points  there are several unresolved issues. First, there is a wide variety of algorithms that use the eigenvectors
Results 1  10
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186,369