A Multilinear Singular Value Decomposition

@article{Lathauwer2000AMS,
  title={A Multilinear Singular Value Decomposition},
  author={Lieven De Lathauwer and Bart De Moor and Joos Vandewalle},
  journal={SIAM J. Matrix Anal. Appl.},
  year={2000},
  volume={21},
  pages={1253-1278}
}
We discuss a multilinear generalization of the singular value decomposition. There is a strong analogy between several properties of the matrix and the higher-order tensor decomposition; uniqueness, link with the matrix eigenvalue decomposition, first-order perturbation effects, etc., are analyzed. We investigate how tensor symmetries affect the decomposition and propose a multilinear generalization of the symmetric eigenvalue decomposition for pair-wise symmetric tensors. 
View via Publisher
lirias.kuleuven.be
3,609 Citations
Highly Influential Citations
543
Background Citations
1,104
Methods Citations
1,307
Results Citations
26

Figures from this paper

3,609 Citations

Citation Type

Citation Type

Singular values and eigenvalues of tensors: a variational approach
  • Lek-Heng Lim
  • Mathematics
    1st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, 2005.
  • 2005
We propose a theory of eigenvalues, eigenvectors, singular values, and singular vectors for tensors based on a constrained variational approach much like the Rayleigh quotient for symmetric matrix…
A locally convergent Jacobi iteration for the tensor singular value problem
TLDR
Local convergence of a parallelizable numerical method (based on the Jacobi iteration) for obtaining the singular values and singular vectors of a tensor is shown.
MODEL REDUCTION THROUGH MULTILINEAR SINGULAR VALUE DECOMPOSITIONS
TLDR
The results in this paper are primarily motvated by the problem to find optimal projection spaces for model reduction purposes, and it is shown that the approximation results outperform earlier singular value based techniques of lower rank approximations of tensors.
Singular Value Decompositions and Low Rank Approximations of Tensors
TLDR
A notion of singular values for arbitrary multilinear mappings is developed and bounds on the error between a tensor and its optimal lower rank approximation are provided.
On the Nuclear Norm and the Singular Value Decomposition of Tensors
  • H. Derksen
  • Mathematics, Computer Science
    Found. Comput. Math.
  • 2016
TLDR
Inspired by the heuristics of convex relaxation, the nuclear norm is considered instead of the rank of a tensor and thenuclear norm of various tensors of interest is determined.
Singular vectors of orthogonally decomposable tensors
TLDR
The spectral theory of orthogonally decomposable tensors is studied and a description of its singular vector tuples as a variety in a product of projective spaces is given.
Singular value decompositions and low rank approximations of multi-linear functionals
TLDR
A notion of singular values for arbitrary multi-linear mappings is introduced and an upperbound is derived on the error between a tensor and its optimal lower rank approximation and a conceptual algorithm is proposed to compute singular value decompositions of tensors.
A Generalized Spectral Theory for Tensors Technical Report
TLDR
The proposed factorization decomposes a symmetric tensor into a product of an orthogonal and a diagonal tensor and offers an expansion of a tensor as a summation of lower rank tensors that are obtained through an outer product defined on matrices.
Iterative Methods for Symmetric Outer Product Tensor Decompositions
We study the symmetric outer product decomposition which decomposes a fully (partially) symmetric tensor into a sum of rank-one fully (partially) symmetric tensors. We present iterative algorithms…
An eigenvalue problem for even order tensors with its applications
TLDR
Using the matrix unfolding of even order tensors, the relationship between a tensor eigenvalue problem and a multilevel matrix eigen value problem is established and it is shown that higher order singular value decomposition of a tensors are the square root of the eigenvalues of the product of the tensor and its conjugate transpose.
...
...

References

SHOWING 1-10 OF 54 REFERENCES
Super-symmetric decomposition of the fourth-order cumulant tensor. Blind identification of more sources than sensors
  • J. Cardoso
  • Physics
    [Proceedings] ICASSP 91: 1991 International Conference on Acoustics, Speech, and Signal Processing
  • 1991
TLDR
Ideas for higher-order array processing are introduced, focusing on fourth-order cumulant statistics, and it is shown that, when dealing with 4-index quantities, symmetries are related to rank properties.
Blind source separation by simultaneous third-order tensor diagonalization
TLDR
It will be shown that, in a Jacobi-type iteration scheme, the computation of an elementary rotation can be reformulated in terms of a simultaneous matrix diagonalization.
Decomposition of quantics in sums of powers of linear forms
Eigen-structure of the fourth-order cumulant tensor with application to the blind source separation problem
  • J. Cardoso
  • Computer Science
    International Conference on Acoustics, Speech, and Signal Processing
  • 1990
TLDR
A special (index-free) tensor formalism to express fourth-order multivariate statistics is proposed, and a quadricovariance tensor which contains theFourth-order joint cumulants is defined, allowing direct extension of many standard second-order method to fourth- order.
FOURTH-ORDER CUMULANT STRUCTURE FORCING
In blind array processing, the array manifold is unknown but, under the signal independence assumption, the signal parameters can be estimated by resorting to higher-order information. We consider…
Tensor Rank is NP-Complete
We prove that computing the rank of a three-dimensional tensor over any finite field is NP-complete. Over the rational numbers the problem is NP-hard.
On the Use of the Singular Value Decomposition in Identification and Signal Processing
TLDR
The OSVD has become a fundamental tool for the formulation and derivation of new concepts such as angles between subspaces, oriented signal-to-signal ratio’s, canonical correlation analysis,... and for the reliable computation of the solutions to problems.
Tensor Methods in Statistics
This book provides a systematic development of tensor methods in statistics, beginning with the study of multivariate moments and cumulants. The effect on moment arrays and on cumulant arrays of…
Independent component analysis based on higher-order statistics only
Most conventional techniques for independent component analysis (or blind source separation) resort to second-order statistics to decorrelate the observed data. The prewhitening step makes these…
Research methods for multimode data analysis
...
...