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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
The Betti numbers of forests
, 2005
"... Abstract. This paper produces a recursive formula of the Betti numbers of certain StanleyReisner ideals (graph ideals associated to forests). This gives a purely combinatorial definition of the projective dimension of these ideals, which turns out to be a new numerical invariant of forests. Finally ..."
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Cited by 15 (0 self)
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Abstract. This paper produces a recursive formula of the Betti numbers of certain StanleyReisner ideals (graph ideals associated to forests). This gives a purely combinatorial definition of the projective dimension of these ideals, which turns out to be a new numerical invariant of forests. Finally, we propose a possible extension of this invariant to general graphs. 0.
Introduction to modular towers: generalizing dihedral group–modular curve connections
 Recent Developments in the Inverse Galois Problem, Cont. Math., proceedings of AMSNSF
, 1995
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Arithmetically CohenMacaulay bundles on threefold hypersurfaces, preprint math.AG/0611620
"... Abstract. We prove that any rank two arithmetically CohenMacaulay vector bundle on a general hypersurface of degree at least six in P 4 must be split. 1. ..."
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Abstract. We prove that any rank two arithmetically CohenMacaulay vector bundle on a general hypersurface of degree at least six in P 4 must be split. 1.
Systems of Algebraic Equations
 In Symbolic and Algebraic Computation. Springer LNCS 72
, 1979
"... Abstract. Let f1; : : : ; fk be k multivariate polynomials which have a finite number of common zeros in the algebraic closure of the ground field, counting the common zeros at infinity. An algorithm is given and proved which reduces the computations of these zeros to the resolution of a single univ ..."
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Abstract. Let f1; : : : ; fk be k multivariate polynomials which have a finite number of common zeros in the algebraic closure of the ground field, counting the common zeros at infinity. An algorithm is given and proved which reduces the computations of these zeros to the resolution of a single univariate equation whose degree is the number of common zeros. This algorithm gives the whole algebraic and geometric structure of the set of zeros (multiplicities, conjugate zeros,...). When all the polynomials have the same degree, the complexity of this algorithm is polynomial relative to the generic number of solutions. 1.
Unramified abelian extensions of Galois covers
 Proceedings of Symposia in Pure Mathematics, Part 1 49
, 1989
"... Abstract: We consider a ramified Galois cover ϕ: ˆ X →P 1 x of the Riemann sphere P 1 x, with monodromy group G. The monodromy group over P 1 x of the maximal unramified abelian exponent n cover of ˆ X is an extension n ˜ G of G by the group (Z/nZ) 2g, where g is the genus of ˆ X. Denote the set of ..."
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Cited by 2 (2 self)
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Abstract: We consider a ramified Galois cover ϕ: ˆ X →P 1 x of the Riemann sphere P 1 x, with monodromy group G. The monodromy group over P 1 x of the maximal unramified abelian exponent n cover of ˆ X is an extension n ˜ G of G by the group (Z/nZ) 2g, where g is the genus of ˆ X. Denote the set of linear equivalence classes of divisors of degree k on ˆ X by Pic k ( ˆ X) = Pic k. This is equipped with a natural G action. We show that the equivalence class of the extension n ˜ G → G is determined by the element of H 1 (G, Pic 0) representing Pic 1 (§2.2). From this we give an effective criterion (involving the Schur multiplier of G) to decide when this group extension splits for all n (§4.2). In particular we easily produce examples from this of cases where ˆ X has G invariant divisor classes of degree 1, but no G invariant divisor of degree 1 (§5.1). The extension n ˜ G → G naturally factors into a sequence n ˜ G → H → G where H is the smallest quotient of n ˜ G giving a frattini cover (§1.1) that fits between n ˜ G and G. Extension of the main result of §4.2 would consider all maximal quotients M of n ˜ G such that M → G splits. We note that the sequence including such an M factors through H, and by example we demonstrate that such maximal quotients M may not be unique (§5.2). INTRODUCTION: We consider a ramified Galois cover ϕ: ˆ X →P 1 x of the Riemann sphere P 1 x, with monodromy group G. Choose an integer n>1. Let ˆ Xn be the maximal unramified abelian exponent n
EXTERIOR ALGEBRAS AND TWO CONJECTURES ON FINITE ABELIAN GROUPS
, 2008
"... Let G be a finite abelian group with G > 1. Let a1,..., ak be k distinct elements of G and let b1,..., bk be (not necessarily distinct) elements of G, where k is a positive integer smaller than the smallest prime divisor of G. We show that there is a permutation π on {1,..., k} such that a1b ..."
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Cited by 2 (0 self)
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Let G be a finite abelian group with G > 1. Let a1,..., ak be k distinct elements of G and let b1,..., bk be (not necessarily distinct) elements of G, where k is a positive integer smaller than the smallest prime divisor of G. We show that there is a permutation π on {1,..., k} such that a1b π(1),..., akb π(k) are distinct, provided that any other prime divisor of G  (if there is any) is greater than k!. This in particular confirms the DasguptaKárolyiSerraSzegedy conjecture for abelian pgroups. We also pose a new conjecture involving determinants and characters, and show that its validity implies Snevily’s conjecture for abelian groups of odd order. Our methods involve exterior algebras and characters.
PARAMETRIZATION OF COSSERAT EQUATIONS
, 902
"... As a matter of fact, the solution space of many systems of ordinary differential (OD) or partial differential (PD) equations in engineering or mathematical physics ”can/cannot ” be parametrized by a certain number of arbitrary functions behaving like ”potentials”. In view of the explicit examples to ..."
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As a matter of fact, the solution space of many systems of ordinary differential (OD) or partial differential (PD) equations in engineering or mathematical physics ”can/cannot ” be parametrized by a certain number of arbitrary functions behaving like ”potentials”. In view of the explicit examples to be met later on, it must be noticed that the parametrizing operator, though often of the