The tensor Golub–Kahan–Tikhonov method applied to the solution of ill‐posed problems with a t‐product structure

Abstract

This paper discusses an application of partial tensor Golub-Kahan bidiagonalization to the solution of large-scale linear discrete ill-posed problems based on the t-product formalism for third-order tensors proposed by Kilmer and Martin (M. E. Kilmer and C. D. Martin, Factorization strategies for third order tensors, Linear Algebra Appl., 435 (2011), pp. 641-658). The solution methods presented first reduce a given (large-scale) problem to a problem of small size by application of a few steps of tensor Golub-Kahan bidiagonalization and then regularize the reduced problem by Tikhonov's method. The regularization operator is a third-order tensor, and the data may be represented by a matrix, that is, a tensor slice, or by a general third-order tensor. A regularization parameter is determined by the discrepancy principle. This results in fully automatic solution methods that neither require a user to choose the number of bidiagonalization steps nor the regularization parameter. The methods presented extend available methods for the solution for linear discrete ill-posed problems defined by a matrix operator to linear discrete ill-posed problems defined by a third-order tensor operator. An interlacing property of singular tubes for third-order tensors is shown and applied. Several algorithms are presented. Computed examples illustrate the advantage of the tensor t-product approach, in comparison with solution methods that are based on matricization of the tensor equation. K E Y W O R D Sdiscrepancy principle, discrete ill-posed problem, t-product, tensor Golub-Kahan bidiagonalization, Tikhonov regularization 1 where  = [a ijk ] 𝓁,m,n i,j,k=1 ∈ R 𝓁×m×n is a third-order tensor, ⃗  ∈ R m×1×n and ⃗  ∈ R 𝓁×1×n are lateral slices of third-order tensors and may be thought of as laterally oriented matrices. The operation * denotes the tensor t-product introduced in the seminal work by Kilmer and Martin 1 and applied to image deblurring problems by Kilmer et al. 1,2 The t-product between

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

“…The solution of linear systems A * X = B, has been recently investigated in literature; see, e.g., [16,17,25,26,27,28], with significant research efforts devoted to the solution of large-scale least squares problems, min…”

“…We are concerned with the solution of (1.1) when the third order tensor A = [a ijk ] ,m,n i,j,k=1 is of ill-determined tubal rank, and the Frobenius norm of the singular tubes of A decay rapidly to zero with increasing index; see, e.g., [25,26]. The singular tubes of A are analogues of the singular values of a matrix, and there are many nonvanishing singular tubes of tiny Frobenius norm of different orders of magnitude close to zero.…”

“…The operator * described below denotes the t-product introduced in the seminal work by Kilmer and Martin [17]. This product has become ubiquitous in tensor literature applications; see, e.g., facial recognition [13], tomographic image reconstruction [24], video completion [32], image classification [21], and image deblurring [8,16,17,25,26,27,28]. Throughout this paper, A F denotes the Frobenius norm of a third order tensor A.…”

“…An advantage of preserving tensor structures when solving (1.1) is to avoid information loss inherent due to flattening; see [16]. Computed examples presented in [25,26,27] illustrate the merits of tensorizing over matricizing or vectorizing ill-posed tensor equations. Solution methods described by El Guide et al [8] involve flattening, i.e., they reduce (1.1) to an equivalent equation involving a matrix and a vector.…”

“…Solution methods described by El Guide et al [8] involve flattening, i.e., they reduce (1.1) to an equivalent equation involving a matrix and a vector. Structure preserving and other techniques for regularizing (1.1) by Tikhonov's approach with the t-product are described and compared in [25,26,27,28]. We will discuss regularization of (1.1) by truncated iterations with the tensor eigenvalue decomposition (tEVD), tSVD, randomized tSVD (R-tSVD), t-product Golub-Kahan bidiagonalization (tGKB) process, and t-product Lanczos (t-Lanczos) process.…”