A Newton–Grassmann Method for Computing the Best Multilinear Rank-$(r_1,$ $r_2,$ $r_3)$ Approximation of a Tensor

2009 ◽  
Vol 31 (2) ◽  
pp. 248-271 ◽  
Author(s):  
Lars Eldén ◽  
Berkant Savas
Keyword(s):  
Multilinear Rank

scholarly journals A Splitting Augmented Lagrangian Method for Low Multilinear-Rank Tensor Recovery

2015 ◽  
Vol 32 (01) ◽  
pp. 1540008 ◽  
Author(s):  
Lei Yang ◽  
Zheng-Hai Huang ◽  
Yu-Fan Li
Keyword(s):  
Augmented Lagrangian ◽  
Augmented Lagrangian Method ◽  
Convex Relaxation ◽  
Linear Constraints ◽  
Lagrangian Method ◽  
Splitting Technique ◽  
Completion Problem ◽  
Rank Tensor ◽  
Tensor Recovery ◽  
Multilinear Rank

This paper studies a recovery task of finding a low multilinear-rank tensor that fulfills some linear constraints in the general settings, which has many applications in computer vision and graphics. This problem is named as the low multilinear-rank tensor recovery problem. The variable splitting technique and convex relaxation technique are used to transform this problem into a tractable constrained optimization problem. Considering the favorable structure of the problem, we develop a splitting augmented Lagrangian method (SALM) to solve the resulting problem. The proposed algorithm is easily implemented and its convergence can be proved under some conditions. Some preliminary numerical results on randomly generated and real completion problems show that the proposed algorithm is very effective and robust for tackling the low multilinear-rank tensor completion problem.

scholarly journals Low Multilinear Rank Approximation of Tensors and Application in Missing Traffic Data

2014 ◽  
Vol 6 ◽  
pp. 157597 ◽  
Author(s):  
Huachun Tan ◽  
Jianshuai Feng ◽  
Zhengdong Chen ◽  
Fan Yang ◽  
Wuhong Wang
Keyword(s):  
Missing Data ◽  
Gradient Descent ◽  
Tensor Completion ◽  
Volume Data ◽  
Real World Application ◽  
Gradient Descent Algorithm ◽  
Bibliographic Data ◽  
Multilinear Rank ◽  
Rank Approximation ◽  
Weighted Optimization

The problem of missing data in multiway arrays (i.e., tensors) is common in many fields such as bibliographic data analysis, image processing, and computer vision. We consider the problems of approximating a tensor by another tensor with low multilinear rank in the presence of missing data and possibly reconstructing it (i.e., tensor completion). In this paper, we propose a weighted Tucker model which models only the known elements for capturing the latent structure of the data and reconstructing the missing elements. To treat the nonuniqueness of the proposed weighted Tucker model, a novel gradient descent algorithm based on a Grassmann manifold, which is termed Tucker weighted optimization (Tucker-Wopt), is proposed for guaranteeing the global convergence to a local minimum of the problem. Based on extensive experiments, Tucker-Wopt is shown to successfully reconstruct tensors with noise and up to 95% missing data. Furthermore, the experiments on traffic flow volume data demonstrate the usefulness of our algorithm on real-world application.

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