# 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

### 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.

### Certifying the Global Optimality of Quartic Minimization over the Sphere

- Computer ScienceJournal 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

- 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.

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

- Computer Science, MathematicsAdvances in Computational Mathematics
- 2023

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

- 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.

## References

SHOWING 1-10 OF 75 REFERENCES

### The Epsilon-Alternating Least Squares for Orthogonal Low-Rank Tensor Approximation and Its Global Convergence

- Computer ScienceSIAM J. Matrix Anal. Appl.
- 2020

The epsilon alternating least squares ($\epsilon$-ALS) is developed and analyzed for canonical polyadic decomposition (approximation) of a higher-order tensor where one or more of the factor matrices…

### Orthogonal Low Rank Tensor Approximation: Alternating Least Squares Method and Its Global Convergence

- Computer ScienceSIAM J. Matrix Anal. Appl.
- 2015

The conventional high-order power method is modified to address the desirable orthogonality via the polar decomposition and it is shown that for almost all tensors the orthogonal alternating least squares method converges globally.

### On the Global Convergence of the Alternating Least Squares Method for Rank-One Approximation to Generic Tensors

- Computer ScienceSIAM J. Matrix Anal. Appl.
- 2014

This paper partially addresses the missing piece by showing that for almost all tensors, the iterates generated by the alternating least squares method for the rank-one approximation converge globally.

### Jacobi-type algorithm for low rank orthogonal approximation of symmetric tensors and its convergence analysis

- Computer Science, MathematicsArXiv
- 2019

A Jacobi-type algorithm to solve the low rank orthogonal approximation problem of symmetric tensors is proposed, and it is proved that an accumulation point is the unique limit point under some conditions.

### Successive Rank-One Approximations for Nearly Orthogonally Decomposable Symmetric Tensors

- Computer ScienceSIAM J. Matrix Anal. Appl.
- 2015

This paper shows that even in the presence of perturbation, SROA can still robustly recover the symmetric canonical decomposition of the underlying tensor.

### Convergence rate analysis for the higher order power method in best rank one approximations of tensors

- MathematicsNumerische Mathematik
- 2018

It is established that the sequence generated by HOPM always converges globally and R-linearly for orthogonally decomposable tensors with order at least 3, and for almost all tensors, all the singular vector tuples are nondegenerate, and so, the HopM “typically” exhibits global R-linear convergence rate.

### Globally convergent Jacobi-type algorithms for simultaneous orthogonal symmetric tensor diagonalization

- Mathematics, Computer ScienceSIAM J. Matrix Anal. Appl.
- 2018

This paper considers a family of Jacobi-type algorithms for a simultaneous orthogonal diagonalization problem of symmetric tensors and proposes and proves a newJacobi-based algorithm in the general setting and proves its global convergence for sufficiently smooth functions.

### Canonical Polyadic Decomposition with a Columnwise Orthonormal Factor Matrix

- MathematicsSIAM J. Matrix Anal. Appl.
- 2012

Orthogonality-constrained versions of the CPD methods based on simultaneous matrix diagonalization and alternating least squares are presented and a simple proof of the existence of the optimal low-rank approximation of a tensor in the case that a factor matrix is columnwise orthonormal is given.

### Local Convergence of the Alternating Least Squares Algorithm for Canonical Tensor Approximation

- MathematicsSIAM J. Matrix Anal. Appl.
- 2012

A local convergence theorem for calculating canonical low-rank tensor approximations (PARAFAC, CANDECOMP) by the alternating least squares algorithm is established. The main assumption is that the…

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

- Mathematics, Computer ScienceSIAM J. Matrix Anal. Appl.
- 2008

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.