Tensor Krylov subspace methods with an invertible linear transform product applied to image processing

Reichel, Lothar; Ugwu, Ugochukwu O.

doi:10.1016/j.apnum.2021.04.007

Cited by 18 publications

(11 citation statements)

References 19 publications

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“…In this subsection, we briefly review some concepts and notations, which play a central role for the elaboration of the tensor global iterative methods based on the c-product; see [7,8] for more details on the c-product. Let A ∈ R n 1 ×n 2 ×n 3 be a real valued third-order tensor, then the operations mat and its inverse ten are defined by…”

Section: Definitions and Properties Of The Cosine Productmentioning

confidence: 99%

A multidimensional principal component analysis via the c-product Golub-Kahan-SVD for classification and face recognition

Hached¹,

Jbilou²,

Koukouvinos³

et al. 2021

Preprint

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Face recognition and identification is a very important application in machine learning. Due to the increasing amount of available data, traditional approaches based on matricization and matrix PCA methods can be difficult to implement. Moreover, the tensorial approaches are a natural choice, due to the mere structure of the databases, for example in the case of color images. Nevertheless, even though various authors proposed factorization strategies for tensors, the size of the considered tensors can pose some serious issues. When only a few features are needed to construct the projection space, there is no need to compute a SVD on the whole data. Two versions of the tensor Golub-Kahan algorithm are considered in this manuscript, as an alternative to the classical use of the tensor SVD which is based on truncated strategies. In this paper, we consider the Tensor Tubal Golub Kahan Principal Component Analysis method which purpose it to extract the main features of images using the tensor singular value decomposition (SVD) based on the tensor cosine product that uses the discrete cosine transform. This approach is applied for classification and face recognition and numerical tests show its effectiveness.

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Section: Definitions and Properties Of The Cosine Productmentioning

confidence: 99%

A multidimensional principal component analysis via the c-product Golub-Kahan-SVD for classification and face recognition

Hached¹,

Jbilou²,

Koukouvinos³

et al. 2021

Preprint

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“…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…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Ugwu¹,

Reichel²

2021

Preprint

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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.

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A tensor bidiagonalization method for higher‐order singular value decomposition with applications

El Hachimi,

Jbilou,

Ratnani

et al. 2023

Numerical Linear Algebra App

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The need to know a few singular triplets associated with the largest singular values of a third‐order tensor arises in data compression and extraction. This paper describes a new method for their computation using the t‐product. Methods for determining a couple of singular triplets associated with the smallest singular values also are presented. The proposed methods generalize available restarted Lanczos bidiagonalization methods for computing a few of the largest or smallest singular triplets of a matrix. The methods of this paper use Ritz and harmonic Ritz lateral slices to determine accurate approximations of the largest and smallest singular triplets, respectively. Computed examples show applications to data compression and face recognition.

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Tensor Krylov subspace methods with an invertible linear transform product applied to image processing

Cited by 18 publications

References 19 publications

A multidimensional principal component analysis via the c-product Golub-Kahan-SVD for classification and face recognition

A multidimensional principal component analysis via the c-product Golub-Kahan-SVD for classification and face recognition

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

A tensor bidiagonalization method for higher‐order singular value decomposition with applications

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