Randomized Quaternion QLP Decomposition for Low-Rank Approximation

Ren, Huan; Ma, Ru-Ru; Liu, Qiaohua; Bai, Zheng-Jian

doi:10.1007/s10915-022-01917-5

Cited by 6 publications

(2 citation statements)

References 42 publications

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“…The complexity of quaternion products motivates the study of fast algorithms for quaternion matrix decomposition. In recent years, by virtue of the real structure-preserving algorithms, many excellent results have been obtained on quaternion LU decomposition, quaternion singular value decomposition, and eigenvalue decomposition of quaternion Hermitian matrices [9][10][11][12][13][14][15][16][17][18]. Similarly, commutative quaternion gives rise to a multitude of operations depending on the composition.…”

Section: Introductionmentioning

confidence: 99%

Algebraic method for LU decomposition in commutative quaternion based on semi‐tensor product of matrices and application to strict image authentication

Ding,

Li,

Liu

et al. 2024

Math Methods in App Sciences

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As a kind of commutative and associative four‐dimensional algebra, commutative quaternion has better applications in color image and signal processing than quaternion. Matrix decomposition is of great concern in the theoretical study and numerical calculation of commutative quaternion. Two kinds of disadvantages of commutative quaternion make the decomposition of commutative quaternion matrix extremely difficult. On one hand, commutative quaternion is not a kind of complete four dimensional division algebra because of the zero divisors. On the other hand, computing the inverse of commutative quaternion is very complicated. In this paper, the semi‐tensor product (STP) of matrices will be used to overcome the above two kinds of shortcomings. And we will propose a real structure‐preserving algorithm based on STP of matrices for commutative quaternion LU decomposition, which makes full use of high‐level operations, relation of operations between commutative quaternion matrices and their ‐representation matrices. Numerical experiments will be provided to demonstrate the efficiency of the real structure‐preserving algorithm based on STP of matrices. Meanwhile, we will apply the structure‐preserving algorithm for strict image authentication.

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Section: Introductionmentioning

confidence: 99%

Algebraic method for LU decomposition in commutative quaternion based on semi‐tensor product of matrices and application to strict image authentication

Ding,

Li,

Liu

et al. 2024

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…In addition to rank minimization, recent studies have utilized quaternion matrix decomposition and randomized techniques to improve the performance of LRQA [17,27,28]. For instance, Miao et al [29] suggested the matrix factorization for the target quaternion matrix, followed by three quaternion-based bilinear factor matrix norm factorization methods for low-rank quaternion matrix completion, which can avoid expensive calculations.…”

Section: Introductionmentioning

confidence: 99%

Efficient quaternion CUR method for low-rank approximation to quaternion matrix

Wu,

Kou,

Cai

et al. 2024

Preprint

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The low-rank quaternion matrix approximation has been successfully applied in many applications involving signal processing and color image processing. However, the cost of quaternion models for generating low-rank quaternion matrix approximation is sometimes considerable due to the computation of the quaternion singular value decomposition (QSVD), which limits their application to real large-scale data. To address this deficiency, an efficient quaternion matrix CUR (QMCUR) method for low-rank approximation is suggested , which provides significant acceleration in color image processing. We first explore the QMCUR approximation method, which uses actual columns and rows of the given quaternion matrix, instead of the costly QSVD. Additionally, two different sampling strategies are used to sample the above-selected columns and rows. Then, the perturbation analysis is performed on the QMCUR approximation of noisy versions of low-rank quater-nion matrices. Extensive experiments on both synthetic and real data further reveal the superiority of the proposed algorithm compared with other algorithms for getting low-rank approximation, in terms of both efficiency and accuracy.

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Randomized block Krylov subspace algorithms for low-rank quaternion matrix approximations

Li,

Liu,

et al. 2023

Numer Algor

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Randomized Quaternion QLP Decomposition for Low-Rank Approximation

Cited by 6 publications

References 42 publications

Algebraic method for LU decomposition in commutative quaternion based on semi‐tensor product of matrices and application to strict image authentication

Algebraic method for LU decomposition in commutative quaternion based on semi‐tensor product of matrices and application to strict image authentication

Efficient quaternion CUR method for low-rank approximation to quaternion matrix

Randomized block Krylov subspace algorithms for low-rank quaternion matrix approximations

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