Tensor Decompositions and Applications

  title={Tensor Decompositions and Applications},
  author={Tamara G. Kolda and Brett W. Bader},
  journal={SIAM Rev.},
This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or $N$-way array. Decompositions of higher-order tensors (i.e., $N$-way arrays with $N \geq 3$) have applications in psycho-metrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Two particular tensor decompositions can be considered to be… 
An Optimization Approach for Fitting Canonical Tensor Decompositions.
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A scalable optimization approach for fitting canonical tensor decompositions
The mathematical calculation of the derivatives of the canonical tensor decomposition is discussed and it is shown that they can be computed efficiently, at the same cost as one iteration of ALS, which is more accurate than ALS and faster than NLS in terms of total computation time.
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Sequential Unfolding SVD for Tensors With Applications in Array Signal Processing
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Multilinear operators for higher-order decompositions
  • T. Kolda
  • Mathematics, Computer Science
  • 2006
Two new multilinear operators are proposed for expressing the matrix compositions that are needed in the Tucker and PARAFAC (CANDECOMP) decompositions and one of them is shorthand for performing an n-mode matrix multiplication for every mode of a given tensor.
Independent component analysis and (simultaneous) third-order tensor diagonalization
It is shown that simultaneous optimal diagonalization of "third-order tensor slices" of the fourth-order cumulant is a suitable strategy and is similar in spirit to the efficient JADE-algorithm.
Multilinear subspace analysis of image ensembles
A dimensionality reduction algorithm that enables subspace analysis within the multilinear framework, based on a tensor decomposition known as the N-mode SVD, the natural extension to tensors of the conventional matrix singular value decomposition (SVD).
Linear image coding for regression and classification using the tensor-rank principle
  • A. Shashua, Anat Levin
  • Computer Science
    Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001
  • 2001
It is found that for regression the tensor-rank coding, as a dimensionality reduction technique, significantly outperforms other techniques like PCA.
A Jacobi-Type Method for Computing Orthogonal Tensor Decompositions
An algorithm for tensors of the form A that is an extension of the Jacobi SVD algorithm for matrices is proposed that is to “condense” a tensor in fewer nonzero entries using orthogonal transformations.