The LSQR method for solving tensor least-squares problems

Electronic Transactions on Numerical Analysis

article

Abdeslem H. Bentbib¹  Asmaa Khouia²  Hassane Sadok³ 

[1] LAMAI, University Cadi Ayyaad;LAMAI, University Cadi Ayyaad, Marrakech, Morocco. LMPA, University Littoral Cote d’Opale Calais;LMPA, University Littoral Cote d’Opale Calais

关键词: HOSVD;    CP decomposition;    color image restoration;    video restoration;    LSQR;   

DOI  :  10.1553/etna_vol55s92

学科分类：数学（综合）

来源: Kent State University * Institute of Computational Mathematics

【 摘 要 】

In this paper, we are interested in finding an approximate solution $ \hat{\mathcal{X}} $ of the tensor least-squares minimization problem $ \min_{\mathcal{X}}\left\|\mathcal{X}\times_1A^{(1)}\times_2A^{(2)}\times_3\cdots\times_NA^{(N)}-\mathcal{G}\right\|$, where $ \mathcal{G}\in \mathbb{R}^{J_1\times J_2\times \cdots \times J_N}$ and $ A^{(i)}\in \mathbb{R}^{J_i\times I_i} $ ($ i=1,\ldots,N $) are known and $ \mathcal{X}\in \mathbb{R}^{I_1\times I_2\times \cdots \times I_N} $ is the unknown tensor to be approximated. Our approach is based on two steps. Firstly, we apply the CP or HOSVD decomposition to the right-hand side tensor $ \mathcal{G} $. Secondly, we perform the well-known Golub-Kahan bidiagonalization for each coefficient matrix $ A^{(i)} $($ i=1,\ldots, N $) to obtain a reduced tensor least-squares minimization problem. This type of equations may appear in color image and video restorations as we described below. Some numerical tests are performed to show the effectiveness of our proposed method.

【 授权许可】

Unknown