Randomized regularized extended Kaczmarz algorithms for tensor recovery

Du, Kui; Sun, Xuejun

doi:10.48550/arxiv.2112.08566

Cited by 4 publications

(6 citation statements)

References 20 publications

(34 reference statements)

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“…26 Recently, Du and Sun extended the matrix randomized extended Kaczmarz method to inconsistent tensor recovery problems. 27 As we know, the MRK method is a popular iterative method for solving large-scale matrix linear systems, that is, the case for l = 1 and p = 1 in the problem (1), and it has wide developments; see for example, References 28-34. Most of these methods can be unified into the sketch-and-project (MSP) method and its adaptive variants proposed by Gower et al 35,36 Inspired by the above research, we propose the tensor sketch-and-project (TSP) method and its adaptive variants to solve the problem (1), followed by their theoretical guarantees.…”

Section: Introductionmentioning

confidence: 99%

“…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%

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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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Section: Introductionmentioning

confidence: 99%

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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“…Later, this method was applied to tensor recovery problems [31]. Recently, Du and Sun extended the matrix randomized extended Kaczmarz method to the inconsistent tensor recovery problems [32].…”

Section: Introductionmentioning

confidence: 99%

“…Besides the randomized algorithms in [28,31,32] mentioned above, there are some research based on random sketching technique for T-product; see e.g., [20,42,43]. In these works, some sketching tensors including the ones extracted from random sampling are formed.…”

Section: Introductionmentioning

confidence: 99%

Sketch-and-project methods for tensor linear systems

TANG¹,

Yajie²,

Zhang³

et al. 2022

Preprint

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We first extend the famous sketch-and-project method and its adaptive variants for matrix linear systems to tensor linear systems with respect to the popular T-product of tensors. Their Fourier domain versions are also investigated. Then, considering that the existing sketching tensor or the way for sampling has some limitations, we propose two improved strategies. Convergence analysis for all 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.

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“…Here (•) † denotes the Moore-Penrose pseudoinverse. We mention that similar algorithms for tensor recovery problems have been given in [4,7].…”

Section: Introductionmentioning

confidence: 99%

Regularized randomized iterative algorithms for factorized linear systems

Du¹

2022

Preprint

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Randomized iterative algorithms for solving a factorized linear system, ABx = b with A ∈ R m× , B ∈ R ×n , and b ∈ R m , have recently been proposed. They take advantage of the factorized form and avoid forming the matrix C = AB explicitly. However, they can only find the minimum norm (least squares) solution. In contrast, the regularized randomized Kaczmarz (RRK) algorithm can find solutions with certain structures from consistent linear systems. In this work, by combining the randomized Kaczmarz algorithm or the randomized Gauss-Seidel algorithm with the RRK algorithm, we propose two novel regularized randomized iterative algorithms to find (least squares) solutions with certain structures of ABx = b. We prove linear convergence of the new algorithms. Computed examples are given to illustrate that the new algorithms can find sparse (least squares) solutions of ABx = b and can be better than the existing randomized iterative algorithms for the corresponding full linear system Cx = b with C = AB.

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Randomized regularized extended Kaczmarz algorithms for tensor recovery

Cited by 4 publications

References 20 publications

Sketch‐and‐project methods for tensor linear systems

Sketch‐and‐project methods for tensor linear systems

Sketch-and-project methods for tensor linear systems

Regularized randomized iterative algorithms for factorized linear systems

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