Low Rank Tensor Decompositions and Approximations

@article{Nie2022LowRT,
  title={Low Rank Tensor Decompositions and Approximations},
  author={Jiawang Nie and L. xilinx Wang and Zequn Zheng},
  journal={ArXiv},
  year={2022},
  volume={abs/2208.07477}
}
. There exist linear relations among tensor entries of low rank ten- sors. These linear relations can be expressed by multi-linear polynomials, which are called generating polynomials. We use generating polynomials to compute tensor rank decompositions and low rank tensor approximations. We prove that this gives a quasi-optimal low rank tensor approximation if the given tensor is sufficiently close to a low rank one. 

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References

SHOWING 1-10 OF 43 REFERENCES

Low Rank Symmetric Tensor Approximations

  • Jiawang Nie
  • Computer Science, Mathematics
    SIAM J. Matrix Anal. Appl.
  • 2017
If the symmetric tensor to be approximated is sufficiently close to a low rank one, it is shown that the computed low rank approximations are quasi-optimal.

Nearly Low Rank Tensors and Their Approximations

This paper studies the LRTAP when the tensor to be approximated is close to a low rank one and proposes a new approach that can be applied to efficiently compute low rank tensor decompositions, especially for large scale tensors.

On the Best Rank-1 and Rank-(

In this paper we discuss a multilinear generalization of the best rank-R approximation problem for matrices, namely, the approximation of a given higher-order tensor, in an optimal leastsquares

Normal Forms for Tensor Rank Decomposition

A new algorithm for computing the tensor rank decomposition or canonical polyadic decomposition of higher-order tensors subject to a rank and genericity constraint is proposed and described in terms of the multigraded regularity of a multihomogeneous ideal.

On the Best Rank-1 and Rank-(R1 , R2, ... , RN) Approximation of Higher-Order Tensors

A multilinear generalization of the best rank-R approximation problem for matrices, namely, the approximation of a given higher-order tensor, in an optimal least-squares sense, by a tensor that has prespecified column rank value, rowRank value, etc.

Generating Polynomials and Symmetric Tensor Decompositions

  • Jiawang Nie
  • Computer Science, Mathematics
    Found. Comput. Math.
  • 2017
This paper characterize the sets of such generating matrices and investigate their properties (e.g., the existence, dimensions, nondefectiveness), and proposes methods for computing symmetric tensor decompositions.

A Link between the Canonical Decomposition in Multilinear Algebra and Simultaneous Matrix Diagonalization

This paper derives a new and relatively weak deterministic sufficient condition for uniqueness in the decomposition of higher-order tensors which have the property that the rank is smaller than the greatest dimension.

Learning Diagonal Gaussian Mixture Models and Incomplete Tensor Decompositions

When the first and third order moments are sufficiently accurate, it is shown that the obtained parameters for the Gaussian mixture models are also highly accurate.

Canonical Polyadic Decomposition of Third-Order Tensors: Reduction to Generalized Eigenvalue Decomposition

This paper presents an algebraic algorithm for the computation of the CPD in cases where none of the factor matrices has full column rank and shows that if the famous Kruskal condition holds, then the C PD can be found algebraically.

Tensor Rank and the Ill-Posedness of the Best Low-Rank Approximation Problem

It is argued 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 rank-r approximations, and a natural way of overcoming the ill-posedness of the low-rank approximation problem is proposed by using weak solutions when true solutions do not exist.