Semidefinite Relaxations for Best Rank-1 Tensor Approximations

Nie, Jiawang; Wang, Li

doi:10.1137/130935112

Cited by 118 publications

(128 citation statements)

References 32 publications

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“…Recently, there are two approaches [17,22] based on semidefinite programming relaxation for finding the global optimal solution to the best rank-1 tensor approximation. A lot of numerical results in [17,22] suggest that both of these two relaxations are very likely to be tight.…”

Section: The Equivalence Between Two Sdp Relaxation Methodsmentioning

confidence: 99%

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A note on semidefinite programming relaxations for polynomial optimization over a single sphere

Jiang

Liu

et al. 2016

Sci. China Math.

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We study two instances of polynomial optimization problem over a single sphere. The first problem is to compute the best rank-1 tensor approximation. We show the equivalence between two recent semidefinite relaxations methods. The other one arises from Bose-Einstein condensates (BEC), whose objective function is a summation of a probably nonconvex quadratic function and a quartic term. These two polynomial optimization problems are closely connected since the BEC problem can be viewed as a structured fourth-order best rank-1 tensor approximation. We show that the BEC problem is NP-hard and propose a semidefinite relaxation with both deterministic and randomized rounding procedures. Explicit approximation ratios for these rounding procedures are presented. The performance of these semidefinite relaxations are illustrated on a few preliminary numerical experiments. Keywordspolynomial optimization over a single sphere, semidefinite programming, best rank-1 tensor approximation, Bose-Einstein condensates MSC(2010) 65K05, 90C22, 90C26 Citation: Hu J, Jiang B, Liu X, et al. A note on semidefinite programming relaxations for polynomial optimization over a single sphere.

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Section: The Equivalence Between Two Sdp Relaxation Methodsmentioning

confidence: 99%

“…A lot of numerical results in [17,22] suggest that both of these two relaxations are very likely to be tight. In this section, we review them and establish their equivalence.…”

Section: The Equivalence Between Two Sdp Relaxation Methodsmentioning

confidence: 99%

A note on semidefinite programming relaxations for polynomial optimization over a single sphere

Jiang

Liu

et al. 2016

Sci. China Math.

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“…In Table 2, we apply our algorithm bbr to find the best rank-1 and rank-2 tensor approximation for symmetric and non symmetric tensors on examples from (Nie and Wang, 2013) and (Ottaviani et al, 2013). For best rank-1 approximation problems with several minimizers (which is the case when there are symmetries), the method proposed in (Nie and Wang, 2013) cannot certify the result and uses a local method to converge to a local extrema.…”

Section: Performancementioning

confidence: 99%

“…For best rank-1 approximation problems with several minimizers (which is the case when there are symmetries), the method proposed in (Nie and Wang, 2013) cannot certify the result and uses a local method to converge to a local extrema. We apply the global border basis relaxation algorithm to find all the minimizers for the best rank 1 approximation problem.…”

Section: Performancementioning

confidence: 99%

Border basis relaxation for polynomial optimization

Bucero

Mourrain

2016

Journal of Symbolic Computation

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A relaxation method based on border basis reduction which improves the efficiency of Lasserre's approach is proposed to compute the infimum of a polynomial function on a basic closed semialgebraic set. A new stopping criterion is given to detect when the relaxation sequence reaches the infimum, using a sparse flat extension criterion. We also provide a new algorithm to reconstruct a finite sum of weighted Dirac measures from a truncated sequence of moments, which can be applied to other sparse reconstruction problems. As an application, we obtain a new algorithm to compute zero-dimensional minimizer ideals and the minimizer points or zero-dimensional G-radical ideals. Experiments show the impact of this new method on significant benchmarks.

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“…For orthogonal third-order tensors (i.e., p = 3), alternating minimization provides polynomial statistical convergence [28]; unfortunately, this guarantee requires using a large number of randomized initializations, and so the results cannot be naturally generalized to higher order tensors without losing polynomial-time computability. Recently, Lassere hierarchy approaches have been proposed for best rank-1 approximations [46] and tensor completion [50]. These have found success on specific numerical examples, but conditions to guarantee global optimality are currently unavailable.…”

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confidence: 99%

Low-Rank Approximation and Completion of Positive Tensors

Aswani¹

2016

SIAM J. Matrix Anal. & Appl.

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Abstract. Unlike the matrix case, computing low-rank approximations of tensors is NP-hard and numerically ill-posed in general. Even the best rank-1 approximation of a tensor is NP-hard. In this paper, we use convex optimization to develop polynomial-time algorithms for low-rank approximation and completion of positive tensors. Our approach is to use algebraic topology to define a new (numerically well-posed) decomposition for positive tensors, which we show is equivalent to the standard tensor decomposition in important cases. Though computing this decomposition is a nonconvex optimization problem, we prove it can be exactly reformulated as a convex optimization problem. This allows us to construct polynomial-time randomized algorithms for computing this decomposition and for solving low-rank tensor approximation problems. Among the consequences is that best rank-1 approximations of positive tensors can be computed in polynomial time. Our framework is next extended to the tensor completion problem, where noisy entries of a tensor are observed and then used to estimate missing entries. We provide a polynomial-time algorithm that for specific cases requires a polynomial (in tensor order) number of measurements, in contrast to existing approaches that require an exponential number of measurements. These algorithms are extended to exploit sparsity in the tensor to reduce the number of measurements needed. We conclude by providing a novel interpretation of statistical regression problems with categorical variables as tensor completion problems, and numerical examples with synthetic data and data from a bioengineered metabolic network show the improved performance of our approach on this problem.

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Semidefinite Relaxations for Best Rank-1 Tensor Approximations

Cited by 118 publications

References 32 publications

A note on semidefinite programming relaxations for polynomial optimization over a single sphere

A note on semidefinite programming relaxations for polynomial optimization over a single sphere

Border basis relaxation for polynomial optimization

Low-Rank Approximation and Completion of Positive Tensors

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