Results 1  10
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12
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 45 (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. 2tensor) 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 3tensor, determining a best rank1 approximation to a 3tensor, determining the rank of a 3tensor over R or C. Hence making tensor computations feasible is likely to be a challenge.
The Euclidean distance degree of an algebraic variety
, 2013
"... The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low rank matrices, the EckartYoung Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest ..."
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Cited by 14 (2 self)
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The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low rank matrices, the EckartYoung Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic geometry. The Euclidean distance degree of a variety is the number of critical points of the squared distance to a generic point outside the variety. Focusing on varieties seen in applications, we present numerous tools for exact computations.
Tensors: a Brief Introduction
, 2014
"... Tensor decompositions are at the core of many Blind Source Separation (BSS) algorithms, either explicitly or implicitly. In particular, the Canonical Polyadic (CP) tensor ..."
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Cited by 11 (3 self)
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Tensor decompositions are at the core of many Blind Source Separation (BSS) algorithms, either explicitly or implicitly. In particular, the Canonical Polyadic (CP) tensor
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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Cited by 5 (1 self)
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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.
On best rank one approximation of tensors
 Journal of Numerical Linear Algebra with Applications
"... All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. ..."
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Cited by 1 (0 self)
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All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately.
COMPUTATIONAL COMPLEXITY OF TENSOR NUCLEAR NORM
"... Abstract. The main result of this paper is that the weak membership problem in the unit ball of a given norm is NPhard if and only if the weak membership problem in the unit ball of the dual norm is NPhard. Equivalently, the approximation of a given norm is polynomial time if and only if the appro ..."
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Abstract. The main result of this paper is that the weak membership problem in the unit ball of a given norm is NPhard if and only if the weak membership problem in the unit ball of the dual norm is NPhard. Equivalently, the approximation of a given norm is polynomial time if and only if the approximation of the dual norm is polynomial time. Using the NPhardness of the approximation of spectral norm of tensors we prove that the approximation of nuclear norm of tensors is NPhard. In addition, we show that bipartite separability of a density matrix is equivalent its corresponding 4tensor having unit nuclear norm, relating these results to quantum information theory. 1.
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"... The number of singular vector tuples and uniqueness of best rank one approximation of tensors Shmuel Friedland ∗ Giorgio Ottaviani† In this paper we discuss the notion of singular vector tuples of a complex valued dmode tensor of dimension m1 ×... × md. We show that a generic tensor has a finite nu ..."
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The number of singular vector tuples and uniqueness of best rank one approximation of tensors Shmuel Friedland ∗ Giorgio Ottaviani† In this paper we discuss the notion of singular vector tuples of a complex valued dmode tensor of dimension m1 ×... × md. We show that a generic tensor has a finite number of singular vector tuples, viewed as points in the corresponding Segre product. We give the formula for the number of singular vector tuples. We show similar results for tensors with partial symmetry. We give analogous results for the homogeneous pencil eigenvalue problem for cubic tensors, i.e. m1 =... = md. We show uniqueness of best approximations for almost all real tensors in the following cases: rank one approximation; rank one approximation for partially symmetric tensors (this approximation is also partially symmetric); rank(r1,..., rd) approximation for dmode tensors.
Lowrank approximation of tensors
, 2014
"... In many applications such as data compression, imaging or genomic data analysis, it is important to approximate a given tensor by a tensor that is sparsely representable. For matrices, i.e. 2tensors, such a representation can be obtained via the singular value decomposition, which allows to compute ..."
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In many applications such as data compression, imaging or genomic data analysis, it is important to approximate a given tensor by a tensor that is sparsely representable. For matrices, i.e. 2tensors, such a representation can be obtained via the singular value decomposition, which allows to compute best rank kapproximations. For very big matrices a low rank approximation using SVD is not computationally feasible. In this case different approximations are available. It seems that variants of CURdecomposition are most suitable. For dmode tensors T ∈ ⊗di=1Rni, with d> 2, many generalizations of the singular value decomposition have been proposed to obtain low tensor rank decompositions. The most appropriate approximation seems to be best (r1,..., rd)approximation, which maximizes the `2 norm of the projection of T on ⊗di=1Ui, where Ui is an ridimensional subspace Rni. One of the most common method is the alternating maximization method (AMM). It is obtained by maximizing on one subspace Ui, while keeping all other fixed, and alternating the procedure repeatedly for i = 1,..., d. Usually, AMM will converge to a local best approximation. This approximation is a fixed point of a corresponding map on Grassmannians. We suggest a Newton method for finding the corresponding fixed point. We also discuss variants of CURapproximation method for tensors. The first part of the paper is a survey on low rank approximation of tensors. The second new part of this paper is a new Newton method for best (r1,..., rd)approximation. We compare numerically different approximation methods.