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52
Most tensor problems are NP hard
- CORR
, 2009
"... The idea that one might extend numerical linear algebra, the collection of matrix computational methods that form the workhorse of scientific and engineering computing, to numerical multilinear algebra, an analogous collection of tools involving hypermatrices/tensors, appears very promising and has ..."
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Cited by 44 (6 self)
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The idea that one might extend numerical linear algebra, the collection of matrix computational methods that form the workhorse of scientific and engineering computing, to numerical multilinear algebra, an analogous collection of tools involving hypermatrices/tensors, appears very promising and has attracted a lot of attention recently. We examine here the computational tractability of some core problems in numerical multilinear algebra. We show that tensor analogues of several standard problems that are readily computable in the matrix (i.e. 2-tensor) case are NP hard. Our list here includes: determining the feasibility of a system of bilinear equations, determining an eigenvalue, a singular value, or the spectral norm of a 3-tensor, determining a best rank-1 approximation to a 3-tensor, determining the rank of a 3-tensor over R or C. Hence making tensor computations feasible is likely to be a challenge.
Finding the largest eigenvalue of a non-negative tensor
- SIAM J. MATRIX ANAL. APPL
, 2009
"... In this paper we propose an iterative method for calculating the largest eigenvalue of an irreducible nonnegative tensor. This method is an extension of a method of Collatz (1942) for calculating the spectral radius of an irreducible nonnegative matrix. Numerical results show that our proposed meth ..."
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Cited by 43 (24 self)
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In this paper we propose an iterative method for calculating the largest eigenvalue of an irreducible nonnegative tensor. This method is an extension of a method of Collatz (1942) for calculating the spectral radius of an irreducible nonnegative matrix. Numerical results show that our proposed method is promising. We also apply the method to studying higher-order Markov chains.
H+-EIGENVALUES OF LAPLACIAN AND SIGNLESS LAPLACIAN TENSORS
, 2014
"... We propose a simple and natural definition for the Laplacian and the signless Lapla-cian tensors of a uniform hypergraph. We study their H+-eigenvalues, i.e., H-eigenvalues with non-negative H-eigenvectors, and H++-eigenvalues, i.e., H-eigenvalues with positive H-eigenvectors. We show that each of ..."
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Cited by 29 (19 self)
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We propose a simple and natural definition for the Laplacian and the signless Lapla-cian tensors of a uniform hypergraph. We study their H+-eigenvalues, i.e., H-eigenvalues with non-negative H-eigenvectors, and H++-eigenvalues, i.e., H-eigenvalues with positive H-eigenvectors. We show that each of the Laplacian tensor, the signless Laplacian tensor, and the adjacency tensor has at most one H++-eigenvalue, but has several other H+-eigenvalues. We identify their largest and small-est H+-eigenvalues, and establish some maximum and minimum properties of these H+-eigenvalues. We then define analytic connectivity of a uniform hypergraph and discuss its application in edge connectivity.
Approximation algorithms for homogeneous polynomial optimization with quadratic constraints
, 2009
"... In this paper, we consider approximation algorithms for optimizing a generic multi-variate homogeneous polynomial function, subject to homogeneous quadratic constraints. Such optimization models have wide applications, e.g., in signal processing, magnetic resonance imaging (MRI), data training, appr ..."
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Cited by 25 (11 self)
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In this paper, we consider approximation algorithms for optimizing a generic multi-variate homogeneous polynomial function, subject to homogeneous quadratic constraints. Such optimization models have wide applications, e.g., in signal processing, magnetic resonance imaging (MRI), data training, approximation theory, and portfolio selection. Since polynomial functions are nonconvex in general, the problems under consideration are all NP-hard. In this paper we shall focus on polynomial-time approximation algorithms. In particular, we first study optimization of a multi-linear tensor function over the Cartesian product of spheres. We shall propose approximation algorithms for such problem and derive worst-case performance ratios, which are shown to depend only on the dimensions of the models. The methods are then extended to optimize a generic multi-variate homogeneous polynomial function with spherical constraints. Likewise, approximation algorithms are proposed with provable relative approximation performance ratios. Furthermore, the constraint set is relaxed to be an intersection of co-centered ellipsoids. In particular, we consider maximization of a homogeneous polynomial over the intersection of ellipsoids centered at the origin, and propose polynomial-time approximation algorithms with provable worst-case performance ratios. Numerical results are reported, illustrating the effectiveness of the approximation algorithms studied.
Maximum block improvement and polynomial optimization
- SIAM Journal on Optimization
"... Abstract. In this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem. The new approach has the following three main ingredients. First, we establish a block coordinate descent type search method for nonlinear opti-mization, with t ..."
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Cited by 16 (5 self)
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Abstract. In this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem. The new approach has the following three main ingredients. First, we establish a block coordinate descent type search method for nonlinear opti-mization, with the novelty being that we only accept a block update that achieves the maximum improvement, hence the name of our new search method: Maximum Block Improvement (MBI). Convergence of the sequence produced by the MBI method to a stationary point is proven. Second, we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem, thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the MBI approach. Third, we propose a scheme to reach a KKT point of the polynomial optimization, provided that a stationary solution for the relaxed tensor problem is avail-able. Numerical experiments have shown that our new method works very efficiently: for a majority of the test instances that we have experimented with, the method finds the global optimal solution at a low computational cost.
LINEAR CONVERGENCE OF THE LZI ALGORITHM FOR WEAKLY POSITIVE TENSORS
, 2012
"... We define weakly positive tensors and study the relations among essentially positive tensors, weakly positive tensors, and primitive tensors. In particular, an explicit linear convergence rate of the Liu-Zhou-Ibrahim(LZI) algorithm for finding the largest eigenvalue of an irreducible nonnegative ten ..."
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Cited by 13 (10 self)
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We define weakly positive tensors and study the relations among essentially positive tensors, weakly positive tensors, and primitive tensors. In particular, an explicit linear convergence rate of the Liu-Zhou-Ibrahim(LZI) algorithm for finding the largest eigenvalue of an irreducible nonnegative tensor, is established for weakly positive tensors. Numerical results are given to demonstrate linear convergence of the LZI algorithm for weakly positive tensors.
Conditions for strong ellipticity of anisotropic elastic materials
- J. Elasticity
"... Abstract In this paper, we derive necessary and sufficient conditions for the strong ellip-ticity condition of anisotropic elastic materials. We first observe that the strong ellipticity condition holds if and only if a second order tensor function is positive definite for any unit vectors. Then we ..."
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Cited by 12 (3 self)
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Abstract In this paper, we derive necessary and sufficient conditions for the strong ellip-ticity condition of anisotropic elastic materials. We first observe that the strong ellipticity condition holds if and only if a second order tensor function is positive definite for any unit vectors. Then we further link this condition to the rank-one positive definiteness of three second-order tensors, three fourth-order tensors and a sixth-order tensor. In particular, we consider conditions of strong ellipticity of the rhombic classes, for which we need to check the copositivity of three second-order tensors and the positive definiteness of a sixth-order tensor. A direct method is presented to verify our conditions.
An Eigenvalue Method for Testing Positive Definiteness of a Multivariate Form
- IEEE Transaction on Automatic Control
, 2008
"... Abstract—In this paper, we present an eigenvalue method for testing positive definiteness of a multivariate form. This problem plays an important role in the stability study of nonlinear au-tonomous systems via Lyapunov’s direct method in automatic control. At first we apply the D’Andrea–Dickenstein ..."
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Cited by 11 (3 self)
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Abstract—In this paper, we present an eigenvalue method for testing positive definiteness of a multivariate form. This problem plays an important role in the stability study of nonlinear au-tonomous systems via Lyapunov’s direct method in automatic control. At first we apply the D’Andrea–Dickenstein version of the classical Macaulay formulas of the resultant to compute the symmetric hyperdeterminant of an even order supersymmetric tensor. By using the supersymmetry property, we give detailed computation procedures for the Bezoutians and specified ordering of monomials in this approach. We then use these formulas to calculate the characteristic polynomial of a fourth order three di-mensional supersymmetric tensor and give an eigenvalue method for testing positive definiteness of a quartic form of three variables. Some numerical results of this method are reported. Index Terms—Eigenvalue method, positive definiteness, super-symmetric tensor, symmetric hyperdeterminant. I.
