Computing Extreme Eigenvalues of Large Scale Hankel Tensors

@article{Chen2016ComputingEE,
  title={Computing Extreme Eigenvalues of Large Scale Hankel Tensors},
  author={Yannan Chen and Liqun Qi and Qun Wang},
  journal={Journal of Scientific Computing},
  year={2016},
  volume={68},
  pages={716-738}
}
Large scale tensors, including large scale Hankel tensors, have many applications in science and engineering. In this paper, we propose an inexact curvilinear search optimization method to compute Z- and H-eigenvalues of mth order n dimensional Hankel tensors, where n is large. Owing to the fast Fourier transform, the computational cost of each iteration of the new method is about $$\mathcal {O}(mn\log (mn))$$O(mnlog(mn)). Using the Cayley transform, we obtain an effective curvilinear search… 
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References

SHOWING 1-10 OF 64 REFERENCES
Fast Hankel tensor–vector product and its application to exponential data fitting
TLDR
A fast algorithm for Hankel tensor–vector products is obtained by embedding a Hankel Tensor into a larger anti-circulant tensor, and the computational complexity is about O(m2nlogmn) for a square Hanko tensor of order m and dimension n.
Shifted Power Method for Computing Tensor Eigenpairs
TLDR
A shifted symmetric higher-order power method (SS-HOPM), which it is shown is guaranteed to converge to a tensor eigenpair, and a fixed point analysis is used to characterize exactly which eigenpairs can and cannot be found by the method.
A sequential subspace projection method for extreme Z-eigenvalues of supersymmetric tensors
TLDR
The main idea of SSPM is to form a 2-dimensional subspace at the current point and then solve the original optimization problem in the subspace and the globalization strategy of random phase can be easily incorporated into S SPM, which promotes the ability to find extreme Z-eigenvalues.
Computing Tensor Eigenvalues via Homotopy Methods
TLDR
An upper bound is derived for the number of equivalence classes of generalized tensor eigenpairs using mixed volume and an algorithm is introduced that combines a heuristic approach and a Newton homotopy method to extract real generalized eigpairs from the found complex generalized eigen pairs.
Efficient MATLAB Computations with Sparse and Factored Tensors
TLDR
This paper considers how specially structured tensors allow for efficient storage and computation, and proposes storing sparse tensors using coordinate format and describes the computational efficiency of this scheme for various mathematical operations, including those typical to tensor decomposition algorithms.
On the Best Rank-1 Approximation of Higher-Order Supersymmetric Tensors
TLDR
It is shown that a symmetric version of the above method converges under assumptions of convexity (or concavity) for the functional induced by the tensor in question, assumptions that are very often satisfied in practical applications.
Z-eigenvalue methods for a global polynomial optimization problem
TLDR
This paper proposes some Z-eigenvalue methods for solving the problem of the best rank-one approximation to higher order tensors, and proposes a direct orthogonal transformation Z- eigenvalue method for this problem in the case of order three and dimension three.
Hankel Tensors: Associated Hankel Matrices and Vandermonde Decomposition
Hankel tensors arise from applications such as signal processing. In this paper, we make an initial study on Hankel tensors. For each Hankel tensor, we associate it with a Hankel matrix and a higher
Hankel tensors, Vandermonde tensors and their positivities☆
Finding the extreme Z-eigenvalues of tensors via a sequential semidefinite programming method
TLDR
An approximation method is proposed to solve the TCLP by using a sequence of semidefinite programming problems, and it gives a numerical algorithm to compute the extreme Z-eigenvalue of an even order tensor with dimension larger than three, which improves the literature.
...
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