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TENSOR RANK AND THE ILLPOSEDNESS OF THE BEST LOWRANK APPROXIMATION PROBLEM
"... There has been continued interest in seeking a theorem describing optimal lowrank approximations to tensors of order 3 or higher, that parallels the Eckart–Young theorem for matrices. In this paper, we argue that the naive approach to this problem is doomed to failure because, unlike matrices, te ..."
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Cited by 193 (13 self)
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There has been continued interest in seeking a theorem describing optimal lowrank approximations to tensors of order 3 or higher, that parallels the Eckart–Young theorem for matrices. In this paper, we argue that the naive approach to this problem is doomed to failure because, unlike matrices, tensors of order 3 or higher can fail to have best rankr approximations. The phenomenon is much more widespread than one might suspect: examples of this failure can be constructed over a wide range of dimensions, orders and ranks, regardless of the choice of norm (or even Brègman divergence). Moreover, we show that in many instances these counterexamples have positive volume: they cannot be regarded as isolated phenomena. In one extreme case, we exhibit a tensor space in which no rank3 tensor has an optimal rank2 approximation. The notable exceptions to this misbehavior are rank1 tensors and order2 tensors (i.e. matrices). In a more positive spirit, we propose a natural way of overcoming the illposedness of the lowrank approximation problem, by using weak solutions when true solutions do not exist. For this to work, it is necessary to characterize the set of weak solutions, and we do this in the case of rank 2, order 3 (in arbitrary dimensions). In our work we emphasize the importance of closely studying concrete lowdimensional examples as a first step towards more general results. To this end, we present a detailed analysis of equivalence classes of 2 × 2 × 2 tensors, and we develop methods for extending results upwards to higher orders and dimensions. Finally, we link our work to existing studies of tensors from an algebraic geometric point of view. The rank of a tensor can in theory be given a semialgebraic description; in other words, can be determined by a system of polynomial inequalities. We study some of these polynomials in cases of interest to us; in particular we make extensive use of the hyperdeterminant ∆ on R 2×2×2.
Symmetric tensors and symmetric tensor rank
 Scientific Computing and Computational Mathematics (SCCM
, 2006
"... Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank1 orderk tensor is the outer product of k nonzero vectors. An ..."
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Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank1 orderk tensor is the outer product of k nonzero vectors. Any symmetric tensor can be decomposed into a linear combination of rank1 tensors, each of them being symmetric or not. The rank of a symmetric tensor is the minimal number of rank1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases, and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz, is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most r is not closed, unless r = 1. Key words. Tensors, multiway arrays, outer product decomposition, symmetric outer product decomposition, candecomp, parafac, tensor rank, symmetric rank, symmetric tensor rank, generic symmetric rank, maximal symmetric rank, quantics AMS subject classifications. 15A03, 15A21, 15A72, 15A69, 15A18 1. Introduction. We
Rank of 3tensors with 2 slices and Kronecker canonical forms, preprint
"... Tensor type data are becoming important recently in various application fields. We determine a rank of a tensor T so that A + T is diagonalizable for a given 3tensor A with 2 slices over the complex and real number field. 1 ..."
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Tensor type data are becoming important recently in various application fields. We determine a rank of a tensor T so that A + T is diagonalizable for a given 3tensor A with 2 slices over the complex and real number field. 1
ELA NATURAL GROUP ACTIONS ON TENSOR PRODUCTS OF THREE REAL VECTOR SPACES WITH FINITELY MANY ORBITS ∗
"... Abstract. Let G be the direct product of the general linear groups of three real vector spaces U, V, W of dimensions l, m, n (2 ≤ l ≤ m ≤ n<∞). Consider the natural action of G on the tensor product of these spaces. The number of Gorbits in X is finite if and only if l =2andm = 2 or 3. In these ..."
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Abstract. Let G be the direct product of the general linear groups of three real vector spaces U, V, W of dimensions l, m, n (2 ≤ l ≤ m ≤ n<∞). Consider the natural action of G on the tensor product of these spaces. The number of Gorbits in X is finite if and only if l =2andm = 2 or 3. In these cases the Gorbits and their connected components are classified, and the closure of each of the components is determined. The proofs make use of recent results of P.G. Parfenov, who solved the same problem for complex vector spaces. Key words. relative invariants. Matrix pencils, KroneckerWeierstrass theory, prehomogeneous vector spaces, AMS subject classifications. 15A69, 15A72 1. Introduction. First we set up some notation. Let U, V, W be real vector spaces of dimension l, m, n respectively, where 2 ≤ l ≤ m ≤ n and denote by U c, V c, W c their complexifications. Set X = U ⊗ V ⊗ W and X c = U c ⊗
c © 2004 Heldermann Verlag Hierarchy of Closures of Matrix Pencils
"... Abstract. The focus of this paper is the standard linear representation of the group SLn(C)×SLm(C)×SL2(C), that is, the tensor product of the corresponding tautological representations. Classification of its orbits is a classic problem, which goes back to the works of Kronecker and Weierstrass. Her ..."
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Abstract. The focus of this paper is the standard linear representation of the group SLn(C)×SLm(C)×SL2(C), that is, the tensor product of the corresponding tautological representations. Classification of its orbits is a classic problem, which goes back to the works of Kronecker and Weierstrass. Here, we summarize some known results about standard linear representations of SLn(C) ×