Linear convergence of an alternating polar decomposition method for low rank orthogonal tensor approximations

@article{Hu2019LinearCO,
  title={Linear convergence of an alternating polar decomposition method for low rank orthogonal tensor approximations},
  author={Shenglong Hu and Ke Ye},
  journal={Mathematical Programming},
  year={2019}
}
  • Shenglong HuKe Ye
  • Published 9 December 2019
  • Mathematics, Computer Science
  • Mathematical Programming
Low rank orthogonal tensor approximation (LROTA) is an important problem in tensor computations and their applications. A classical and widely used algorithm is the alternating polar decomposition method (APD). In this article, an improved version iAPD of the classical APD is proposed. For the first time, all the following four fundamental properties are established for iAPD: (i) the algorithm converges globally and the whole sequence converges to a KKT point without any assumption; (ii) it… 
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7 Citations
Background Citations
2
Methods Citations
4

7 Citations

On Approximation Algorithm for Orthogonal Low-Rank Tensor Approximation

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  • Computer Science
    Journal of Optimization Theory and Applications
  • 2022
The presented results fill a gap left in Yang (SIAM J Matrix Anal Appl 41:1797–1825, 2020), where the approximation bound of that approximation algorithm was established when there is only one orthonormal factor.

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Recovering orthogonal tensors under arbitrarily strong, but locally correlated, noise

The problem of recovering an orthogonally decomposable tensor with a subset of elements distorted by noise with arbitrarily large magnitude can be solved through a system of coupled Sylvester-like equations and how to accelerate their solution by an alternating solver is shown.

Certifying the Global Optimality of Quartic Minimization over the Sphere

  • Shenglong Hu
  • Computer Science
    Journal of the Operations Research Society of China
  • 2021
Two classes of methods which are able to certify the global optimality of the quartic minimization over the sphere are presented, i.e., algebraic methods and semidefinite program (SDP) relaxation methods.

Nondegeneracy of eigenvectors and singular vector tuples of tensors

  • Shenglong Hu
  • Computer Science, Physics
    Science China Mathematics
  • 2022
The main results are: (i) each (Z-)eigenvector/singular vector tuple of a generic tensor is nondegenerate, and (ii) each nonzero Z-eigen vector/ Singular vector tuples of an orthogonally decomposable tensors is nondEGenerate.

Half-quadratic alternating direction method of multipliers for robust orthogonal tensor approximation

This paper derives a robust orthogonal tensor CPD model with Cauchy loss, which is resistant to heavy-tailed noise such as theCauchy noise, or outliers, and develops the so-called half-quadratic alternating direction method of multipliers (HQ-ADMM) to solve the model.

Polar decomposition based algorithms on the product of Stiefel manifolds with applications in tensor approximation

It turns out that well-known algorithms are all special cases of this general algorithmic framework and its symmetric variant, and the convergence results subsume the results found in the literature designed for those special cases.

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