Tensor regularization by truncated iteration: a comparison of some solution methods for large-scale linear discrete ill-posed problem with a t-product

Abstract

This paper describes and compares some structure preserving techniques for the solution of linear discrete ill-posed problems with the t-product. A new randomized tensor singular value decomposition (R-tSVD) with a t-product is presented for low tubal rank tensor approximations. Regularization of linear inverse problems by truncated tensor eigenvalue decomposition (T-tEVD), truncated tSVD (T-tSVD), randomized T-tSVD (RT-tSVD), t-product Golub-Kahan bidiagonalization (tGKB) process, and t-product Lanczos (t-Lanczos) process are considered. A solution method that is based on reusing tensor Krylov subspaces generated by the tGKB process is described. The regularization parameter is the number of iterations required by each method. The discrepancy principle is used to determine this parameter. Solution methods that are based on truncated iterations are compared with solution methods that combine Tikhonov regularization with the tGKB and t-Lanczos processes. Computed examples illustrate the performance of these methods when applied to image and gray-scale video restorations. Our new RT-tSVD method is seen to require less CPU time and yields restorations of higher quality than the T-tSVD method.

Randomized algorithms for fast computation of low rank tensor ring model

“…Note that in Algorithms 6 and 7, we need the tubal rank as input, however, this can be numerically estimated for a given approximation error bound. For example, we can use the randomized fixed-precision developed in [26] and for the case of tensors, the randomized rank-revealing algorithm proposed in [27,28,25] is applicable.…”

Tensor Conjugate Gradient Methods with Automatically Determination of Regularization Parameters for Ill-Posed Problems with t-Product

“…This method exhibits notable advantages in handling large-scale datasets and holds significant potential for applications in image data compression and analysis. Ugwu and Reichel [25] proposed a new random tensor singular value decomposition (R-tSVD), which improves T-tSVD.…”

Tensor Conjugate-Gradient Methods With Automatically Determination of Regularization Parameters for Ill-Posed Problems With T-product

“…The randomized tensor singular value decomposition (rt-SVD) method in [3] was presented for computing super large data sets, and has prospects in image data compression and analysis. Ugwu and Reichel [23] proposed a new random tensor singular value decomposition (R-tSVD), which improves the truncated tensor singular value decomposition (T-tSVD) in [1]. Kilmer et al [2] presented a tensor Conjugate-Gradient (t-CG) method for tensor linear systems A * X = B corresponding to the least-squares problems.…”