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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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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 194 (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.
Singular values and eigenvalues of tensors: a variational approach.
 In Proceedings of the IEEE International Workshop on Computational Advances in MultiSensor Adaptive Processing,
, 2005
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Scalable tensor decompositions for multiaspect data mining
 In ICDM 2008: Proceedings of the 8th IEEE International Conference on Data Mining
, 2008
"... Modern applications such as Internet traffic, telecommunication records, and largescale social networks generate massive amounts of data with multiple aspects and high dimensionalities. Tensors (i.e., multiway arrays) provide a natural representation for such data. Consequently, tensor decompositi ..."
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Cited by 64 (2 self)
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Modern applications such as Internet traffic, telecommunication records, and largescale social networks generate massive amounts of data with multiple aspects and high dimensionalities. Tensors (i.e., multiway arrays) provide a natural representation for such data. Consequently, tensor decompositions such as Tucker become important tools for summarization and analysis. One major challenge is how to deal with highdimensional, sparse data. In other words, how do we compute decompositions of tensors where most of the entries of the tensor are zero. Specialized techniques are needed for computing the Tucker decompositions for sparse tensors because standard algorithms do not account for the sparsity of the data. As a result, a surprising phenomenon is observed by practitioners: Despite the fact that there is enough memory to store both the input tensors and the factorized output tensors, memory overflows occur during the tensor factorization process. To address this intermediate blowup problem, we propose MemoryEfficient Tucker (MET). Based on the available memory, MET adaptively selects the right execution strategy during the decomposition. We provide quantitative and qualitative evaluation of MET on real tensors. It achieves over 1000X space reduction without sacrificing speed; it also allows us to work with much larger tensors that were too big to handle before. Finally, we demonstrate a data mining casestudy using MET. 1
TensorTextures: Multilinear ImageBased Rendering
 ACM TRANSACTIONS ON GRAPHICS
, 2004
"... This paper introduces a tensor framework for imagebased rendering. In particular, we develop an algorithm called TensorTextures that learns a parsimonious model of the bidirectional texture function (BTF) from observational data. Given an ensemble of images of a textured surface, our nonlinear, gen ..."
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Cited by 60 (0 self)
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This paper introduces a tensor framework for imagebased rendering. In particular, we develop an algorithm called TensorTextures that learns a parsimonious model of the bidirectional texture function (BTF) from observational data. Given an ensemble of images of a textured surface, our nonlinear, generative model explicitly represents the multifactor interaction implicit in the detailed appearance of the surface under varying photometric angles, including local (pertexel) reflectance, complex mesostructural selfocclusion, interreflection and selfshadowing, and other BTFrelevant phenomena. Mathematically, TensorTextures is based on multilinear algebra, the algebra of higherorder tensors, hence its name. It is computed through a decomposition known as the Nmode SVD, an extension to tensors of the conventional matrix singular value decomposition (SVD). We demonstrate the application of TensorTextures to the imagebased rendering of natural and synthetic textured surfaces under continuously varying viewpoint and illumination conditions.
A NEWTONGRASSMANN METHOD FOR COMPUTING THE BEST MULTILINEAR RANK(R_1, R_2, R_3) APPROXIMATION OF A Tensor
"... We derive a Newton method for computing the best rank(r_1, r_2, r_3) approximation of a given J × K × L tensor A. The problem is formulated as an approximation problem on a product of Grassmann manifolds. Incorporating the manifold structure into Newton’s method ensures that all iterates generated ..."
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Cited by 35 (8 self)
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We derive a Newton method for computing the best rank(r_1, r_2, r_3) approximation of a given J × K × L tensor A. The problem is formulated as an approximation problem on a product of Grassmann manifolds. Incorporating the manifold structure into Newton’s method ensures that all iterates generated by the algorithm are points on the Grassmann manifolds. We also introduce a consistent notation for matricizing a tensor, for contracted tensor products and some tensoralgebraic manipulations, which simplify the derivation of the Newton equations and enable straightforward algorithmic implementation. Experiments show a quadratic convergence rate for the NewtonGrassmann algorithm.
Matlab tensor classes for fast algorithm prototyping
 ACM Trans. Math. Software
, 2004
"... Sandia is a multiprogram laboratory operated by Sandia Corporation, ..."
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Cited by 34 (4 self)
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Sandia is a multiprogram laboratory operated by Sandia Corporation,
BiQuadratic Optimization over Unit Spheres and Semidefinite Programming Relaxations
, 2008
"... Abstract. This paper studies the socalled biquadratic optimization over unit spheres min x∈R n,y∈R m bijklxiyjxkyl ..."
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Cited by 32 (15 self)
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Abstract. This paper studies the socalled biquadratic optimization over unit spheres min x∈R n,y∈R m bijklxiyjxkyl
Handwritten Digit Classification using Higher Order Singular Value Decomposition
"... In this paper we present two algorithms for handwritten digit classification based on the higher order singular value decomposition (HOSVD). The first algorithm uses HOSVD for construction of the class models and achieves classification results with error rate lower than 6%. The second algorithm use ..."
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Cited by 23 (2 self)
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In this paper we present two algorithms for handwritten digit classification based on the higher order singular value decomposition (HOSVD). The first algorithm uses HOSVD for construction of the class models and achieves classification results with error rate lower than 6%. The second algorithm uses the HOSVD for tensor approximation simultaneously in two modes. Classification results for the second algorithm are almost down at 5 % even though the approximation reduces the original training data with more than 98 % before the construction of the class models. The actual classification in the test phase for both algorithms is conducted by solving a series least squares problems. Considering computational amount for the test presented the second algorithm is twice as efficient as the first one.