# 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} }

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…

## 7 Citations

### Half-Quadratic Alternating Direction Method of Multipliers for Robust Orthogonal Tensor Approximation

- Computer Science, Mathematics
- 2020

This paper derives a robust orthogonal tensor CPD model with Cauchy loss, which is resistant to heavy-tailed noise or outliers and shows that the whole sequence generated by the algorithm globally converges to a stationary point of the problem under consideration.

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

- Computer Science, MathematicsArXiv
- 2019

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.

### On Approximation Algorithm for Orthogonal Low-Rank Tensor Approximation

- Computer ScienceJournal 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.

### Jacobi-type algorithms for homogeneous polynomial optimization on Stiefel manifolds with applications to tensor approximations

- Mathematics, Computer ScienceArXiv
- 2021

This paper studies the gradient based Jacobi-type algorithms to maximize two classes of homogeneous polynomials with orthogonality constraints, and establishes their convergence properties, and proposes theJacobi-GP and Jacobi -MGP algorithms, and establish their global convergence without any further condition.

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

- Computer Science, MathematicsNumerical Linear Algebra with Applications
- 2022

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.

### Nondegeneracy of eigenvectors and singular vector tuples of tensors

- Computer Science, PhysicsScience 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.

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