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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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Cited by 101 (22 self)
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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
ON THE IDEALS OF SECANT VARIETIES OF SEGRE VARIETIES
, 2003
"... We establish basic techniques for studying the ideals of secant varieties of Segre varieties. We solve a conjecture of Garcia, Stillman and Sturmfels on the generators of the ideal of the first secant variety in the case of three factors and solve the conjecture settheoretically for an arbitrary n ..."
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Cited by 66 (13 self)
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We establish basic techniques for studying the ideals of secant varieties of Segre varieties. We solve a conjecture of Garcia, Stillman and Sturmfels on the generators of the ideal of the first secant variety in the case of three factors and solve the conjecture settheoretically for an arbitrary number of factors. We determine the low degree components of the ideals of secant varieties of small dimension in a few cases.
On the ideals of Secant Varieties to certain rational varieties
, 2006
"... If X ⊂ Pn is a reduced and irreducible projective variety, it is interesting to find the equations describing the (higher) secant varieties of X. In this paper we find those equations in the following cases: • X = Pn1 ×... × Pnt × Pn is the Segre embedding of the product and n is “large ” with res ..."
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Cited by 31 (2 self)
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If X ⊂ Pn is a reduced and irreducible projective variety, it is interesting to find the equations describing the (higher) secant varieties of X. In this paper we find those equations in the following cases: • X = Pn1 ×... × Pnt × Pn is the Segre embedding of the product and n is “large ” with respect to the ni (Theorem 2.4); • X is a SegreVeronese embedding of some products with 2 or three factors; • X is a Del Pezzo surface.
When is the Fourier transform of an elementary function elementary
 Selecta Math. (N.S
"... Let F be a local field, ψ a nontrivial unitary additive character of F, and V a finite dimensional vector space over F. Let us say that a complex function on V is elementary if it has the form k∏ g(x) = Cψ(Q(x)) χj(Pj(x)), x ∈ V, j=1 where C ∈ C, Q a rational function (the phase function), Pj are p ..."
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Cited by 25 (4 self)
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Let F be a local field, ψ a nontrivial unitary additive character of F, and V a finite dimensional vector space over F. Let us say that a complex function on V is elementary if it has the form k∏ g(x) = Cψ(Q(x)) χj(Pj(x)), x ∈ V, j=1 where C ∈ C, Q a rational function (the phase function), Pj are polynomials, and χj multiplicative characters of F. For generic χj, this function canonically extends to a distribution on V (if char(F)=0). Occasionally, the Fourier transform of an elementary function is also an elementary function (the basic example is the Gaussian integral: k = 0, Q is a nondegenerate quadratic form). It is interesting to determine when exactly this happens. This question is the main subject of our study. In the first part of this paper, we show that for F = R or C, if the Fourier transform of an elementary function g ̸ = 0 with phase function −Q such that detd 2 Q ̸ = 0 is another elementary function g ∗ with phase function Q ∗ , then Q ∗ is the Legendre transform of Q (the “semiclassical condition”). We study properties and examples of phase functions satisfying this condition, and give a classification of phase functions such that both Q and Q ∗ are of the form f(x)/t, where f is a homogeneous cubic polynomial and t is an additional variable (this is one of the simplest possible situations). Unexpectedly, the proof uses Zak’s classification theorem for Severi varieties. In the second part of the paper, we give a necessary and sufficient condition for an elementary function to have an elementary Fourier transform (in an appropriate “weak ” sense) and explicit formulas for such Fourier transforms in the case when Q and Pj are monomials, over any local field F. We also describe a generalization of these results to the case of monomials of norms of finite extensions of F. Finally, we generalize some of the above results (including Fourier integration formulas) to the case when F = C and Q comes from a prehomogeneous vector space. 1.