The LSQR method for solving tensor least-squares problems

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 $) known \mathcal{X}\in \mathbb{R}^{I_1\times I_2\times I_N} is unknown to be approximated. Our approach based on two steps. Firstly, apply CP or HOSVD decomposition right-hand side \mathcal{G} $. Secondly, perform well-known Golub-Kahan bidiagonalization for each coefficient matrix A^{(i)} $($ i=1,\ldots, N obtain a reduced problem. This type equations may appear color image video restorations as described below. Some numerical tests performed show effectiveness our proposed method.