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42 results on '"Liu, Qiaohua"'

6. A contribution to condition numbers of the multidimensional total least squares problem with linear equality constraint

Author

Liu, Qiaohua, Jia, Zhigang, and Wei, Yimin

Subjects
Mathematics - Numerical Analysis, 65F35
Abstract

This paper is devoted to condition numbers of the multidimensional total least squares problem with linear equality constraint (TLSE). Based on the perturbation theory of invariant subspace, the TLSE problem is proved to be equivalent to a multidimensional unconstrained weighed total least squares problem in the limit sense. With a limit technique, Kronecker-product-based formulae for normwise, mixed and componentwise condition numbers of the minimum Frobenius norm TLSE solution are given. Compact upper bounds of these condition numbers are provided to reduce the storage and computation cost. All expressions and upper bounds of these condition numbers unify the ones for the single-dimensional TLSE problem and multidimensional total least squares problem. Some numerical experiments are performed to illustrate our results., Comment: 27 pages

Published
2020

7. Condition numbers of the mixed least squares-total least squares problem: revisited

Author

Liu, Qiaohua, Zhang, Qian, and Shen, Dongmei

Subjects
Mathematics - Numerical Analysis, 65F35, 65F20
Abstract

A new closed formula for the first order perturbation estimate of the mixed least squares-total least squares (MTLS) solution is presented. It is mathematically equivalent to the one by Zheng and Yang(Numer. Linear Algebra Appl. 2019; 26(4):e2239). With this formula, general and structured normwise, mixed and componentwise condition numbers of the MTLS problem are derived. Perturbation bounds based on the normwise condition number, and compact forms for the upper bounds of mixed and componentwise condition numbers are also given in order for economic storage and efficient computation. It is shown that the condition numbers and perturbation bound of the TLS problem are unified in the ones of the MTLS problem., Comment: 20 pages

Published
2020

8. Randomized Quaternion Singular Value Decomposition for Low-Rank Approximation

Author

Liu, Qiaohua, Ling, Sitao, and Jia, Zhigang

Subjects
Mathematics - Numerical Analysis, 65F55, G.1.3
Abstract

This paper presents a randomized quaternion singular value decomposition (QSVD) algorithm for low-rank matrix approximation problems, which are widely used in color face recognition, video compression, and signal processing problems. With quaternion normal distribution based random sampling, the randomized QSVD algorithm projects a high-dimensional data to a low-dimensional subspace and then identifies an approximate range subspace of the quaternion matrix. The key statistical properties of quaternion Wishart distribution are proposed and used to perform the approximation error analysis of the algorithm. Theoretical results show that the randomized QSVD algorithm can trace dominant singular value decomposition triplets of a quaternion matrix with acceptable accuracy. Numerical experiments also indicate the rationality of proposed theories. Applied to color face recognition problems, the randomized QSVD algorithm obtains higher recognition accuracies and behaves more efficient than the known Lanczos-based partial QSVD and a quaternion version of fast frequent directions algorithm., Comment: 31 pages, 9 figures

Published
2020

9. On condition numbers of the total least squares problem with linear equality constraint

Author

Liu, Qiaohua and Jia, Zhigang

Subjects
Mathematics - Numerical Analysis, 65F35
Abstract

This paper is devoted to condition numbers of the total least squares problem with linear equality constraint (TLSE). With novel limit techniques, closed formulae for normwise, mixed and componentwise condition numbers of the TLSE problem are derived. Computable expressions and upper bounds for these condition numbers are also given to avoid the costly Kronecker product-based operations. The results unify the ones for the TLS problem. For TLSE problems with equilibratory input data, numerical experiments illustrate that normwise condition number-based estimate is sharp to evaluate the forward error of the solution, while for sparse and badly scaled matrices, mixed and componentwise condition numbers-based estimates are much tighter., Comment: 19 pages

Published
2020

14. On rank‐revealing QR factorizations of quaternion matrices.

Author

Liu, Qiaohua and Li, Chuge

Subjects
*IMAGE compression, *SIGNAL denoising, *QUATERNIONS, *MATRIX decomposition, *FACTORIZATION, *GREEDY algorithms
Abstract

This work develops theories and algorithms for computing rank‐revealing QR factorizations (qRRQR) of quaternion matrices. First, by introducing a novel quaternion determinant, a quasi‐Cramer's rule is established to investigate the existence theory of the qRRQR factorization of an m×n$$ m\times n $$ quaternion matrix A$$ \boldsymbol{A} $$. The proposed theory provides a systematic approach for selecting k$$ k $$ linearly independent columns of A$$ \boldsymbol{A} $$ in order to ensure that the resulting diagonal blocks R11$$ {\boldsymbol{R}}_{11} $$, R22$$ {\boldsymbol{R}}_{22} $$ of the R‐factor possess rank revealing properties in both spectral and Frobenius norms. Secondly, by increasing the quaternion determinants of R11$$ {\boldsymbol{R}}_{11} $$ in each iteration by a factor of at least f−1$$ {f}^{-1} $$(0

Published
2024
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21. Condition numbers of the mixed least squares-total least squares problem revisited.

Author

Liu, Qiaohua, Zhang, Qian, and Shen, Dongmei

Subjects
*LINEAR algebra, *NUMBER theory
Abstract

A recent study on the condition numbers of the mixed least squares-total least squares (MTLS) problem is due to Zheng and Yang (Numer Linear Algebra Appl. 2019;26(4):e2239). However, the associated expressions are not compact and the Kronecker-product operations make the computation costly. In this paper, we first present new and alternative closed formula for the first order perturbation estimate and condition numbers of the MTLS solution. Then we reveal the relationship between the new formula and Zheng and Yang's result. Several new computable formulae and perturbation bounds for the normwise condition number of the MTLS solution are also provided. Finally, mixed and componentwise condition numbers, structured condition numbers are investigated. Through a number of tests, they are shown to be tighter than the normwise condition numbers for sparse and structured problems. [ABSTRACT FROM AUTHOR]

Published
2023
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31. Elementary Transformation and its Applications for Split Quaternion Matrices.

Author

Wang, Minghui, Yue, Lingling, and Liu, Qiaohua

Abstract

In this paper, the elementary transformation is discussed for split quaternion matrices and the upper triangulation process is given. Then two new determinants are defined by means of real and complex representation matrices and the sufficient condition for the existence of LU decomposition is obtained. Finally, the inverse of the split quaternion matrix is studied and its existence condition and computation method are given. [ABSTRACT FROM AUTHOR]

Published
2020
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42. The hyperbolic elimination method for solving the equality constrained indefinite least squares problem.

Author

Liu, Qiaohua, Pan, Baozhen, and Wang, Qian

Subjects
*LEAST squares, *PROBLEM solving, *EXPONENTIAL functions, *ELIMINATION (Mathematics), *PERTURBATION theory, *FACTORIZATION, *NUMERICAL analysis, *CONSTRAINT satisfaction
Abstract

Recently, Liu and Wang [Q. Liu and M. Wang, Algebraic properties and perturbation results for the indefinite least squares problem with equality constraints, Int. J. Comp. Math. 87(2) (2010), pp. 425-434.] proved that the solution of the equality constrained indefinite least squares (ILSE) problem min Bx=d(b-Ax)TJ(b-Ax), J=diag(-Iq, Ip) is the limit of the solution of the unconstrained weighted indefinite least squares (WILS) problem [image omitted] and J˜=diag(Is, J) as the weight μ tends to infinity, assuming that B has full row rank and xTATJAx>0 for all nonzero x∈null (B). Based on this observation, we derive a type of elimination method by applying the hyperbolic QR factorization method to above WILS problem and taking the limit analytically. Theoretical analysis shows that the method obtained is forward stable under a reasonable assumption. We illustrate our results with numerical tests. [ABSTRACT FROM AUTHOR]

Published
2010
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