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

@article{Wang2014OnTG, title={On the Global Convergence of the Alternating Least Squares Method for Rank-One Approximation to Generic Tensors}, author={Liqi Wang and Moody T. Chu}, journal={SIAM J. Matrix Anal. Appl.}, year={2014}, volume={35}, pages={1058-1072} }

Tensor decomposition has important applications in various disciplines, but it remains an extremely challenging task even to this date. A slightly more manageable endeavor has been to find a low rank approximation in place of the decomposition. Even for this less stringent undertaking, it is an established fact that tensors beyond matrices can fail to have best low rank approximations, with the notable exception that the best rank-one approximation always exists for tensors of any order. Toward…

## 43 Citations

### Convergence of Alternating Least Squares Optimisation for Rank-One Approximation to High Order Tensors

- Computer Science
- 2015

In this analysis, the global convergence and the rate of convergence of the ALS algorithm for the rank-one approximation problem are focused on and it is shown that the ALS method can converge sublinearly, Q-lin Early, and even Q-superlinearly.

### 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.

### Some results concerning rank-one truncated steepest descent directions in tensor spaces

- Computer Science2015 International Conference on Sampling Theory and Applications (SampTA)
- 2015

This work presents a conceptual review of this approach to finding low-rank solutions to matrix or tensor optimization tasks by greedy rank-one methods, and provides some new insights.

### GLOBAL RANK-1 APPROXIMATION FOR ORDER-3 TENSORS

- Computer Science
- 2018

The empirical results of two investigations are reported, finding that the rank-1 approximat on problem can easily have many local solutions and that most of the lower rank approximation methods available in the literature might have severely missed the target.

### Convergence Analysis of Alternating Direction Methods: A General Framework and Its Applications to Tensor Approximations

- Mathematics
- 2018

For problems involving multiple variables, the notion of solving a sequence of simplified problems by fixing all but one variable a time and alternating among the variables has been exploited in a…

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

- Mathematics, Computer ScienceMathematical Programming
- 2022

An improved version iAPD of the classical APD is proposed, which exhibits an overall sublinear convergence with an explicit rate which is sharper than the usual $O(1/k)$ for first order methods in optimization.

### On the convergence of higher-order orthogonal iteration

- Mathematics
- 2015

Abstract The higher-order orthogonal iteration (HOOI) has been popularly used for finding a best low-multilinear rank approximation of a tensor. However, its convergence is still an open question. In…

### Alternating Least Squares as Moving Subspace Correction

- MathematicsSIAM J. Numer. Anal.
- 2018

This work is able to provide an alternative and conceptually simple derivation of the asymptotic convergence rate of the two-sided block power method of numerical algebra for computing the dominant singular subspaces of a rectangular matrix.

### Convergence analysis of an SVD-based algorithm for the best rank-1 tensor approximation

- Computer ScienceLinear Algebra and its Applications
- 2018

### 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.

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