The Best Rank-One Approximation Ratio of a Tensor Space

Qi, Liqun

doi:10.1137/100795802

Cited by 82 publications

(67 citation statements)

References 23 publications

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“…By Theorem 2.1 of this paper and Theorem 3.1, (2.4), and (2.5) of [27], we have the following theorem.…”

Section: An Applicationmentioning

confidence: 69%

“…We show that the best symmetric rank-1 approximation to a symmetric tensor is its best rank-1 approximation, which can be stated as follows. This theorem shows that Conjecture 1 of [27] is true. We prove this theorem by induction on m. For the case that m = 2, Theorem 2.1 is true from the well-known Eckart-Young theorem.…”

Section: Th Entry Ismentioning

confidence: 72%

“…More recently, in [27], the best rank-1 approximation ratio of a tensor space was introduced. A positive lower bound of the best rank-1 approximation ratio of a tensor space gives a convergence rate for the greedy rank-1 update algorithm.…”

Section: Introductionmentioning

confidence: 99%

“…A positive lower bound of the best rank-1 approximation ratio of a tensor space gives a convergence rate for the greedy rank-1 update algorithm. A conjecture (Conjecture 1) was given in [27] that the best symmetric rank-1 approximation to a symmetric tensor is its best rank-1 approximation for m ≥ 4 and any n. If this conjecture is true, then a positive lower bound can be given for the best rank-1 approximation ratio of a symmetric tensor space of order m for m ≥ 2.…”

Section: Introductionmentioning

confidence: 99%

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The Best Rank-1 Approximation of a Symmetric Tensor and Related Spherical Optimization Problems

Zhang¹,

Ling²,

Qi³

2012

SIAM J. Matrix Anal. & Appl.

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Abstract. In this paper, we show that for a symmetric tensor, its best symmetric rank-1 approximation is its best rank-1 approximation. Based on this result, a positive lower bound for the best rank-1 approximation ratio of a symmetric tensor is given. Furthermore, a higher order polynomial spherical optimization problem can be reformulated as a multilinear spherical optimization problem. Then, we present a modified power algorithm for solving the homogeneous polynomial spherical optimization problem. Numerical results are presented, illustrating the effectiveness of the proposed algorithm.Key words. symmetric tensor, the best rank-1 approximation, the best symmetric rank-1 approximation, power algorithm , i 2 , . . . , i m ). Symmetric tensors arise in higher order derivatives of smooth functions, moments, and cumulants of random vectors and have wide applications in signal and image processing, blind source separation (BSS), statistics, investment science, and so on; see [1,8,17,18,26] and references therein.A tensor is said to be rank-1 if it can be expressed as an outer product of a number of vectors. Specifically, if these vectors are all equal, then the tensor is called a symmetric rank-1 tensor. Given an m-order n-dimensional square tensor A, rank-1 tensor B = λx(1) · · · x (m) is said to be its best rank-1 approximation if it minimizes the least-squares cost function A − B F over the manifold of rank-1 tensors. Similarly, symmetric rank-1 tensor C = μx m is said to be the best symmetric rank-1 approximation if it minimizes the least-squares cost function A − C F over the manifold of symmetric rank-1 tensors. From optimization theory, B and C can be obtained by solving the optimization problems

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“…By Theorem 2.1 of this paper and Theorem 3.1, (2.4), and (2.5) of [27], we have the following theorem.…”

Section: An Applicationmentioning

confidence: 69%

Section: Th Entry Ismentioning

confidence: 72%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Best Rank-1 Approximation of a Symmetric Tensor and Related Spherical Optimization Problems

Zhang¹,

Ling²,

Qi³

2012

SIAM J. Matrix Anal. & Appl.

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“…Taking the similar technique of (18) we obtain , when m and n are very large, the bounds in (30) need more computations than the bounds in (10). As stated in Section 1, by the bounds in (30), we can obtain a more sharp bound of min 2R;x2R n ;kxk 2 D1 kA x m k F , which plays an important role in the symmetric best rank-one approximation [12][13][14][15][16]. This can be seen in the following example.…”

Section: Bounds For Weakly Symmetric Nonnegative Tensorsmentioning

confidence: 79%

Bounds for the Z-eigenpair of general nonnegative tensors

Liu

2016

Open Mathematics

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Abstract:In this paper, we consider the Z-eigenpair of a tensor. A lower bound and an upper bound for the Z-spectral radius of a weakly symmetric nonnegative irreducible tensor are presented. Furthermore, upper bounds of Z-spectral radius of nonnegative tensors and general tensors are given. The proposed bounds improve some existing ones. Numerical examples are reported to show the effectiveness of the proposed bounds.

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Finding the extreme Z‐eigenvalues of tensors via a sequential semidefinite programming method

Huang

2013

Numerical Linear Algebra App

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SUMMARYIn this paper, we first introduce the tensor conic linear programming (TCLP), which is a generalization of the space TCLP. Then an approximation method, by using a sequence of semidefinite programming problems, is proposed to solve the TCLP. In particular, we reformulate the extreme Z-eigenvalue problem as a special TCLP. It gives a numerical algorithm to compute the extreme Z-eigenvalue of an even order tensor with dimension larger than three, which improves the literature. Numerical experiments show the efficiency of the proposed method.

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The Best Rank-One Approximation Ratio of a Tensor Space

Cited by 82 publications

References 23 publications

The Best Rank-1 Approximation of a Symmetric Tensor and Related Spherical Optimization Problems

The Best Rank-1 Approximation of a Symmetric Tensor and Related Spherical Optimization Problems

Bounds for the Z-eigenpair of general nonnegative tensors

Finding the extreme Z‐eigenvalues of tensors via a sequential semidefinite programming method

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