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Tensor Decompositions and Applications
 SIAM REVIEW
, 2009
"... This survey provides an overview of higherorder tensor decompositions, their applications, and available software. A tensor is a multidimensional or N way array. Decompositions of higherorder tensors (i.e., N way arrays with N â¥ 3) have applications in psychometrics, chemometrics, signal proce ..."
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Cited by 228 (14 self)
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This survey provides an overview of higherorder tensor decompositions, their applications, and available software. A tensor is a multidimensional or N way array. Decompositions of higherorder tensors (i.e., N way arrays with N â¥ 3) have applications in psychometrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, etc. Two particular tensor decompositions can be considered to be higherorder extensions of the matrix singular value decompo
sition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rankone tensors, and the Tucker decomposition is a higherorder form of principal components analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The Nway Toolbox and Tensor Toolbox, both for MATLAB, and the Multilinear Engine are examples of software packages for working with tensors.
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 75 (10 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.
Geometry and the complexity of matrix multiplication
, 2007
"... Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, ..."
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Cited by 15 (1 self)
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Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.
Geometric complexity theory and tensor rank, arXiv:1011.1350
, 2010
"... Mulmuley and Sohoni [25, 26] proposed to view the permanent versus determinant problem as a specific orbit closure problem and to attack it by methods from geometric invariant and representation theory. We adopt these ideas towards the goal of showing lower bounds on the border rank of specific tens ..."
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Cited by 9 (5 self)
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Mulmuley and Sohoni [25, 26] proposed to view the permanent versus determinant problem as a specific orbit closure problem and to attack it by methods from geometric invariant and representation theory. We adopt these ideas towards the goal of showing lower bounds on the border rank of specific tensors, in particular for matrix multiplication. We thus study specific orbit closure problems for the group G = GL(W1) × GL(W2) × GL(W3) acting on the tensor product W = W1 ⊗ W2 ⊗ W3 of complex finite dimensional vector spaces. Let Gs = SL(W1) × SL(W2) × SL(W3). A key idea from [26] is that the irreducible Gsrepresentations occurring in the coordinate ring of the Gorbit closure of astabletensorw∈Ware exactly those having a nonzero invariant with respect to the stabilizer group of w. However, we prove that by considering Gsrepresentations, only trivial lower bounds on border rank can be shown. It is thus necessary to study Grepresentations, which leads to geometric extension problems that are beyond the scope of the subgroup restriction problems emphasized in [25, 26] and its follow up papers. We prove a very modest lower bound on the border rank of matrix multiplication tensors using Grepresentations. This shows at least that the barrier for Gsrepresentations can be overcome. To advance, we suggest the coarser approach to replace the semigroup of representations of a tensor by its moment polytope. We prove first results towards determining the moment polytopes of matrix multiplication and unit tensors. A full version of this paper is available at arxiv.org/abs/1011.1350
SECANT VARIETIES OF TORIC VARIETIES
, 2005
"... Abstract. Let XP be a smooth projective toric variety of dimension n embedded in P r using all of the lattice points of the polytope P. We compute the dimension and degree of the secant variety Sec XP. We also give explicit formulas in dimensions 2 and 3 and obtain partial results for the projective ..."
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Cited by 5 (0 self)
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Abstract. Let XP be a smooth projective toric variety of dimension n embedded in P r using all of the lattice points of the polytope P. We compute the dimension and degree of the secant variety Sec XP. We also give explicit formulas in dimensions 2 and 3 and obtain partial results for the projective varieties XA embedded using a set of lattice points A ⊂ P ∩Z n containing the vertices of P and their nearest neighbors. 1.
EQUATIONS FOR LOWER BOUNDS ON BORDER RANK
, 1305
"... Abstract. We present new methods for determining polynomials in the ideal of the variety of bilinear maps of border rank at most r. We apply these methods to several cases including the case r = 6 in the space of bilinear maps C 4 × C 4 → C 4. This space of bilinear maps includes the matrix multipli ..."
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Abstract. We present new methods for determining polynomials in the ideal of the variety of bilinear maps of border rank at most r. We apply these methods to several cases including the case r = 6 in the space of bilinear maps C 4 × C 4 → C 4. This space of bilinear maps includes the matrix multiplication operator M2 for two by two matrices. We show these newly obtained polynomials do not vanish on the matrix multiplication operator M2, which gives a new proof that the border rank of the multiplication of 2×2 matrices is seven. Other examples are considered along with an explanation of how to implement the methods. Acknowledgements. We thank Peter Bürgisser for important discussions and suggestions, and the anonymous reviewer for many helpful comments. 1.
ABSTRACT
, 1210
"... We prove the lower bound R(Mm) ≥ 3 2 m2 − 2 on the border rank of m × m matrix multiplication by exhibiting explicit representation theoretic (occurence) obstructions in the sense of Mulmuley and Sohoni’s geometric complexity theory(GCT)program. Whilethisboundisweakerthanthe one recentlyobtained by ..."
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We prove the lower bound R(Mm) ≥ 3 2 m2 − 2 on the border rank of m × m matrix multiplication by exhibiting explicit representation theoretic (occurence) obstructions in the sense of Mulmuley and Sohoni’s geometric complexity theory(GCT)program. Whilethisboundisweakerthanthe one recentlyobtained by Landsbergand Ottaviani, these are the first significant lower bounds obtained within the GCT program. Behind the proof is thenew combinatorial concept of obstruction designs, which encode highest weight vectors in Sym d ⊗ 3 (C
Explicit Lower Bounds via Geometric Complexity Theory
, 1210
"... We prove the lower bound R(Mm) ≥ 3 2 m2 − 2 on the border rank of m × m matrix multiplication by exhibiting explicit representation theoretic (occurence) obstructions in the sense the geometric complexity theory (GCT) program. While this bound is weaker than the one recently obtained by Landsberg a ..."
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We prove the lower bound R(Mm) ≥ 3 2 m2 − 2 on the border rank of m × m matrix multiplication by exhibiting explicit representation theoretic (occurence) obstructions in the sense the geometric complexity theory (GCT) program. While this bound is weaker than the one recently obtained by Landsberg and Ottaviani, these are the first significant lower bounds obtained within the GCT program. Behind the proof is an explicit description of the highest weight vectors in Sym d ⊗ 3 (C n in terms of combinatorial objects, called obstruction designs. This description results from analyzing the process of polarization and SchurWeyl duality.