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Jacobi algorithm for the best low multilinear rank approximation of symmetric tensors
 SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
, 2013
"... The problem discussed in this paper is the symmetric best low multilinear rank approximation of thirdorder symmetric tensors. We propose an algorithm based on Jacobi rotations, for which symmetry is preserved at each iteration. Two numerical examples are provided indicating the need for such algo ..."
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The problem discussed in this paper is the symmetric best low multilinear rank approximation of thirdorder symmetric tensors. We propose an algorithm based on Jacobi rotations, for which symmetry is preserved at each iteration. Two numerical examples are provided indicating the need for such algorithms. An important part of the paper consists of proving that our algorithm converges to stationary points of the objective function. This can be considered an advantage of the proposed algorithm over existing symmetrypreserving algorithms in the literature.
Local minima of the best low multilinear rank approximation of tensors
"... AbstractHigherorder tensors are generalizations of vectors and matrices to thirdor even higherorder arrays of numbers. We consider a generalization of column and row rank of a matrix to tensors, called multilinear rank. Given a higherorder tensor, we are looking for another tensor, as close as ..."
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AbstractHigherorder tensors are generalizations of vectors and matrices to thirdor even higherorder arrays of numbers. We consider a generalization of column and row rank of a matrix to tensors, called multilinear rank. Given a higherorder tensor, we are looking for another tensor, as close as possible to the original one and with multilinear rank bounded by prespecified numbers. In this paper, we give an overview of recent results pertaining the associated cost function. It can have a number of local minima, which need to be interpreted carefully. Convergence to the global minimum cannot be guaranteed with the existing algorithms. We discuss the conclusions that we have drawn from extensive simulations and point out some hidden problems that might occur in real applications.