Randomized Kaczmarz for tensor linear systems

Ma, Anna; Molitor, Denali

doi:10.1007/s10543-021-00877-w

Cited by 25 publications

(16 citation statements)

References 34 publications

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“…The former can be solved by the TRK method [29], which is a specal case of the TSP method [26], while the latter can be solved by the MERK methods, which are specal cases of the MESP method. In this example, we compare the empirical performance of the TERK methods (including the TERK-both, TERK-left and TERK-right methods) for the tensor equation ( 1), the TRK [29] method for the tensor linear system (15), and the MERK methods (including the MERK-both [21,22], MERK-left and MERK-right methods) for the matrix equation ( 16). We generate the tubal matrices A ∈ K m×r l , B ∈ K s×n l and X ∈ K r×s l by using the MATLAB function randn, and construct a tensor equation by setting C = A * X * B.…”

Section: Resultsmentioning

confidence: 99%

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On sketch-and-project methods for solving tensor equations

Tang¹,

Zhang²,

Li³

2022

Preprint

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We first propose the regular sketch-and-project method for solving tensor equations with respect to the popular t-product. Then, three adaptive sampling strategies and three corresponding adaptive sketch-and-project methods are derived. We prove that all the proposed methods have linear convergence in expectation. Furthermore, we investigate the Fourier domain versions and some special cases of the new methods, where the latter corresponds to some existing matrix equation methods. Finally, numerical experiments are presented to demonstrate and test the feasibility and effectiveness of the proposed methods for solving tensor equations.

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Section: Resultsmentioning

confidence: 99%

“…2 ⌉ in line 8 of Algorithm 5 will be the same. As pointed out in [26,29], it would be better to use different sketching matrices for these ⌈ l+1 2 ⌉ independent matrix equations.…”

Section: The Fourier Version Of the Tesp Methodsmentioning

confidence: 99%

On sketch-and-project methods for solving tensor equations

Tang¹,

Zhang²,

Li³

2022

Preprint

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“…However, they have some limitations. For example, the Gaussian random tensor in Reference 37 and 38 is defined as a tensor whose first frontal slice is created by the standard normal distribution and other frontal slices are all zeros; the random sampling tensor in References 15,23,26,27 is formed similarly, that is, its first frontal slice is a sampling matrix but other frontal slices are all zeros. In this way, the transformed tensor by the discrete Fourier transform (DFT) along the third dimension will have the same frontal slices.…”

Section: Introductionmentioning

confidence: 99%

“…To solve the problem (1), Ma and Molitor 23 extended the matrix randomized Kaczmarz (MRK) method 24,25 and called it the tensor randomized Kaczmarz (TRK) method. Later, this method was applied to tensor recovery problems 26 .…”

Section: Introductionmentioning

confidence: 99%

Sketch‐and‐project methods for tensor linear systems

TANG

Yajie

Zhang

et al. 2022

Numerical Linear Algebra App

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For tensor linear systems with respect to the popular t-product, we first present the sketch-and-project method and its adaptive variants. Their Fourier domain versions are also investigated. Then, considering that the existing sketching tensor or way for sampling has some limitations, we propose two improved strategies. Convergence analyses for the methods mentioned above are provided. We compare our methods with the existing ones using synthetic and real data.Numerical results show that they have quite decent performance in terms of the number of iterations and running time.

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“…Miao et al [45] introduced the tensor TLS and discussed the stochastic perturbation bounds for the tensor Moore-Penrose inverse based on the tensor-tensor product (T-product). Tensor T-product is also widely used in the fields of tensor linear systems, the tensor recovery and traffic models [9,33,39]. Besides, the tensor Krylov subspace and Golub-Kahan-Tikhonov methods are used to speed up the computation in real applications [13,15,48,49].…”

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confidence: 99%

Tensor Regularized Total Least Squares Methods with Applications to Image and Video Deblurring

Han¹,

Wei²

2022

Preprint

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Total least squares (TLS) is an effective method for solving linear equations with the situations, when noise is not just in observation matrices but also in mapping matrices. Moreover, the Tikhonov regularization is widely used in plenty of ill-posed problems. In this paper, we extend the regularized total least squares (RTLS) method from the matrix form due to Golub, Hansen and O'Leary, to the tensor form proposing the tensor regularized total least squares (TR-TLS) method for solving ill-conditioned tensor systems of equations. Properties and algorithms about the solution of the TR-TLS problem, which might be similar to those of the RTLS, are also presented and proved. Based on this method, some applications in image and video deblurring are explored. Numerical examples illustrate the TR-TLS, compared with the existing methods.

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Randomized Kaczmarz for tensor linear systems

Cited by 25 publications

References 34 publications

On sketch-and-project methods for solving tensor equations

On sketch-and-project methods for solving tensor equations

Sketch‐and‐project methods for tensor linear systems

Tensor Regularized Total Least Squares Methods with Applications to Image and Video Deblurring

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