The LSQR method for solving tensor least-squares problems

Abstract

In this paper, we are interested in finding an approximate solution X̂ of the tensor least-squares minimization problem minX ∥∥X ×1 A(1) ×2 A(2) ×3 · · · ×N A(N) − G∥∥, where G ∈ RJ1×J2×···×JN and A(i) ∈ RJi×Ii (i = 1, . . . , N ) are known and X ∈ RI1×I2×···×IN 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 G. Secondly, we perform the well-known Golub-Kahan bidiagonalization for each coefficient matrix A(i)(i = 1, . . . , 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.