Jinbao Jian, Qiongxuan Huang, Jianghua Yin, Guodong Ma
This article presents and analyzes a family of three‐term conjugate gradient projection methods with the inertial technique for solving large‐scale nonlinear pseudo‐monotone equations with convex constraints. The generated search direction exhibits good properties independent of line searches. The global convergence of the family is proved without the Lipschitz continuity of the underlying mapping. Furthermore, under the locally Lipschitz continuity assumption, we conduct a thorough analysis related to the asymptotic and non‐asymptotic global convergence rates in terms of iteration complexity. To our knowledge, this is the first iteration‐complexity analysis for inertial gradient‐type projection methods, in the literature, under such a assumption. Numerical experiments demonstrate the computational efficiency of the family, showing its superiority over three existing inertial methods. Finally, we apply the proposed family to solve practical problems such as ‐regularized logistic regression, sparse signal restoration and image restoration problems, highlighting its effectiveness and potential for real‐world applications.
{"title":"A Family of Inertial Three‐Term CGPMs for Large‐Scale Nonlinear Pseudo‐Monotone Equations With Convex Constraints","authors":"Jinbao Jian, Qiongxuan Huang, Jianghua Yin, Guodong Ma","doi":"10.1002/nla.2589","DOIUrl":"https://doi.org/10.1002/nla.2589","url":null,"abstract":"This article presents and analyzes a family of three‐term conjugate gradient projection methods with the inertial technique for solving large‐scale nonlinear pseudo‐monotone equations with convex constraints. The generated search direction exhibits good properties independent of line searches. The global convergence of the family is proved without the Lipschitz continuity of the underlying mapping. Furthermore, under the locally Lipschitz continuity assumption, we conduct a thorough analysis related to the asymptotic and non‐asymptotic global convergence rates in terms of iteration complexity. To our knowledge, this is the first iteration‐complexity analysis for inertial gradient‐type projection methods, in the literature, under such a assumption. Numerical experiments demonstrate the computational efficiency of the family, showing its superiority over three existing inertial methods. Finally, we apply the proposed family to solve practical problems such as ‐regularized logistic regression, sparse signal restoration and image restoration problems, highlighting its effectiveness and potential for real‐world applications.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"10 1","pages":""},"PeriodicalIF":4.3,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142222838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Kabiru Ahmed, Mohammed Yusuf Waziri, Abubakar Sani Halilu, Salisu Murtala, Habibu Abdullahi
The one parameter conjugate gradient method by Hager and Zhang (Pac J Optim, 2(1):35–58, 2006) represents a family of descent iterative methods for solving large‐scale minimization problems. The nonnegative parameter of the scheme determines the weight of conjugacy and descent, and by extension, the numerical performance of the method. The scheme, however, does not converge globally for general nonlinear functions, and when the parameter approaches 0, the scheme reduces to the conjugate gradient method by Hestenes and Stiefel (J Res Nat Bur Stand, 49:409–436, 1952), which in practical sense does not perform well due to the jamming phenomenon. By carrying out eigenvalue analysis of an adaptive two parameter Hager–Zhang type method, a new scheme is presented for system of monotone nonlinear equations with its application in compressed sensing. The proposed scheme was inspired by nice attributes of the Hager–Zhang method and the various schemes designed with double parameters. The scheme is also applicable to nonsmooth nonlinear problems. Using fundamental assumptions, analysis of the global convergence of the scheme is conducted and preliminary report of numerical experiments carried out with the scheme and some recent methods indicate that the scheme is promising.
{"title":"Signal and image reconstruction with a double parameter Hager–Zhang‐type conjugate gradient method for system of nonlinear equations","authors":"Kabiru Ahmed, Mohammed Yusuf Waziri, Abubakar Sani Halilu, Salisu Murtala, Habibu Abdullahi","doi":"10.1002/nla.2583","DOIUrl":"https://doi.org/10.1002/nla.2583","url":null,"abstract":"The one parameter conjugate gradient method by Hager and Zhang (<jats:italic>Pac J Optim</jats:italic>, 2(1):35–58, 2006) represents a family of descent iterative methods for solving large‐scale minimization problems. The nonnegative parameter of the scheme determines the weight of conjugacy and descent, and by extension, the numerical performance of the method. The scheme, however, does not converge globally for general nonlinear functions, and when the parameter approaches 0, the scheme reduces to the conjugate gradient method by Hestenes and Stiefel (<jats:italic>J Res Nat Bur Stand</jats:italic>, 49:409–436, 1952), which in practical sense does not perform well due to the jamming phenomenon. By carrying out eigenvalue analysis of an adaptive two parameter Hager–Zhang type method, a new scheme is presented for system of monotone nonlinear equations with its application in compressed sensing. The proposed scheme was inspired by nice attributes of the Hager–Zhang method and the various schemes designed with double parameters. The scheme is also applicable to nonsmooth nonlinear problems. Using fundamental assumptions, analysis of the global convergence of the scheme is conducted and preliminary report of numerical experiments carried out with the scheme and some recent methods indicate that the scheme is promising.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"18 1","pages":""},"PeriodicalIF":4.3,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142222839","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper studies the superlinear convergence of Krylov iterations under the streamline‐diffusion preconditioning operator for convection‐dominated elliptic problems. First, convergence results are given involving the diffusion parameter . Then the limiting case is studied on the operator level, and the convergence results are extended to this situation under some conditions, in spite of the lack of compactness of the perturbation operators. An explicit rate is also given.
{"title":"Superlinear Krylov convergence under streamline diffusion FEM for convection‐dominated elliptic operators","authors":"János Karátson","doi":"10.1002/nla.2586","DOIUrl":"https://doi.org/10.1002/nla.2586","url":null,"abstract":"This paper studies the superlinear convergence of Krylov iterations under the streamline‐diffusion preconditioning operator for convection‐dominated elliptic problems. First, convergence results are given involving the diffusion parameter . Then the limiting case is studied on the operator level, and the convergence results are extended to this situation under some conditions, in spite of the lack of compactness of the perturbation operators. An explicit rate is also given.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"10 1","pages":""},"PeriodicalIF":4.3,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142222840","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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 quaternion matrix . The proposed theory provides a systematic approach for selecting linearly independent columns of in order to ensure that the resulting diagonal blocks , of the R‐factor possess rank revealing properties in both spectral and Frobenius norms. Secondly, by increasing the quaternion determinants of in each iteration by a factor of at least (), a greedy algorithm with cyclic block pivoting strategy is derived to implement the qRRQR factorization. This algorithm costs about real floating‐point operations, which is as nearly efficient as quaternion QR with column pivoting for most problems. It also shows the effectiveness and reliability, particularly when dealing with a class of quaternion Kahan matrices. Furthermore, two improved greedy algorithms are proposed. Experiments evaluations are conducted on synthetic data as well as color image compression and quaternion signals denoising. The experimental results validate the usefulness and efficiency of our improved algorithm across various scenarios.
这项研究开发了计算四元数矩阵的秩揭示 QR 因式分解（qRRQR）的理论和算法。首先，通过引入一个新颖的四元数行列式，建立了一个准克拉默规则来研究四元数矩阵 qRRQR 因式分解的存在性理论。所提出的理论提供了一种选择四元矩阵线性独立列的系统方法，以确保 R 因子的对角块 , 在谱规范和弗罗贝尼斯规范中都具有秩揭示特性。其次，通过在每次迭代中将四元数行列式至少增加（）的因子，推导出一种具有循环块支点策略的贪婪算法，以实现 qRRQR 因式分解。该算法的浮点运算成本约为实数，对于大多数问题来说，其效率与采用列支点的四元数 QR 算法相当。它还显示了有效性和可靠性，特别是在处理一类四元数 Kahan 矩阵时。此外，还提出了两种改进的贪心算法。对合成数据以及彩色图像压缩和四元数信号去噪进行了实验评估。实验结果验证了我们的改进算法在各种场景下的实用性和效率。
{"title":"On rank‐revealing QR factorizations of quaternion matrices","authors":"Qiaohua Liu, Chuge Li","doi":"10.1002/nla.2585","DOIUrl":"https://doi.org/10.1002/nla.2585","url":null,"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 quaternion matrix . The proposed theory provides a systematic approach for selecting linearly independent columns of in order to ensure that the resulting diagonal blocks , of the <jats:italic>R</jats:italic>‐factor possess rank revealing properties in both spectral and Frobenius norms. Secondly, by increasing the quaternion determinants of in each iteration by a factor of at least (), a greedy algorithm with cyclic block pivoting strategy is derived to implement the qRRQR factorization. This algorithm costs about real floating‐point operations, which is as nearly efficient as quaternion QR with column pivoting for most problems. It also shows the effectiveness and reliability, particularly when dealing with a class of quaternion Kahan matrices. Furthermore, two improved greedy algorithms are proposed. Experiments evaluations are conducted on synthetic data as well as color image compression and quaternion signals denoising. The experimental results validate the usefulness and efficiency of our improved algorithm across various scenarios.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"44 1","pages":""},"PeriodicalIF":4.3,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142222841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we determine probabilistic approximations of the entries of random perturbation matrices implementing the Markoff inequality. These approximations are used to derive with prescribed probability asymptotic componentwise perturbation bounds of some orthogonal and unitary matrix decompositions. We show that the probabilistic asymptotic bounds are significantly less conservative than the corresponding deterministic perturbation bounds. As case studies, we consider the determining of probabilistic perturbation bounds of the QR decomposition, the singular value decomposition and the Schur decomposition of a matrix using an unified method for asymptotic componentwise perturbation analysis of these decompositions. It is demonstrated that the probability bounds of the orthogonal transformations, singular values and eigenvalues are much tighter than the corresponding deterministic asymptotic bounds. The probabilistic bounds derived are appropriate for perturbation analysis of large‐order matrix problems.
在本文中，我们根据马尔科夫不等式确定了随机扰动矩阵项的概率近似值。我们利用这些近似值，以规定概率推导出一些正交和单元矩阵分解的渐近分量扰动边界。我们证明，概率渐近界值的保守性明显低于相应的确定性扰动界值。作为案例研究，我们考虑用一种统一的方法来确定矩阵的 QR 分解、奇异值分解和舒尔分解的概率扰动边界，并对这些分解进行渐近分量扰动分析。结果表明，正交变换、奇异值和特征值的概率边界比相应的确定性渐近边界要严格得多。推导出的概率边界适用于大阶矩阵问题的扰动分析。
{"title":"Probabilistic perturbation bounds of matrix decompositions","authors":"Petko H. Petkov","doi":"10.1002/nla.2582","DOIUrl":"https://doi.org/10.1002/nla.2582","url":null,"abstract":"In this article, we determine probabilistic approximations of the entries of random perturbation matrices implementing the Markoff inequality. These approximations are used to derive with prescribed probability asymptotic componentwise perturbation bounds of some orthogonal and unitary matrix decompositions. We show that the probabilistic asymptotic bounds are significantly less conservative than the corresponding deterministic perturbation bounds. As case studies, we consider the determining of probabilistic perturbation bounds of the QR decomposition, the singular value decomposition and the Schur decomposition of a matrix using an unified method for asymptotic componentwise perturbation analysis of these decompositions. It is demonstrated that the probability bounds of the orthogonal transformations, singular values and eigenvalues are much tighter than the corresponding deterministic asymptotic bounds. The probabilistic bounds derived are appropriate for perturbation analysis of large‐order matrix problems.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"29 1","pages":""},"PeriodicalIF":4.3,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142222845","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we propose, analyze and demonstrate a dynamic momentum method to accelerate power and inverse power iterations with minimal computational overhead. The method can be applied to real diagonalizable matrices, is provably convergent with acceleration in the symmetric case, and does not require a priori spectral knowledge. We review and extend background results on previously developed static momentum accelerations for the power iteration through the connection between the momentum accelerated iteration and the standard power iteration applied to an augmented matrix. We show that the augmented matrix is defective for the optimal parameter choice. We then present our dynamic method which updates the momentum parameter at each iteration based on the Rayleigh quotient and two previous residuals. We present convergence and stability theory for the method by considering a power‐like method consisting of multiplying an initial vector by a sequence of augmented matrices. We demonstrate the developed method on a number of benchmark problems, and see that it outperforms both the power iteration and often the static momentum acceleration with optimal parameter choice. Finally, we present and demonstrate an explicit extension of the algorithm to inverse power iterations.
{"title":"Dynamically accelerating the power iteration with momentum","authors":"Christian Austin, Sara Pollock, Yunrong Zhu","doi":"10.1002/nla.2584","DOIUrl":"https://doi.org/10.1002/nla.2584","url":null,"abstract":"In this article, we propose, analyze and demonstrate a dynamic momentum method to accelerate power and inverse power iterations with minimal computational overhead. The method can be applied to real diagonalizable matrices, is provably convergent with acceleration in the symmetric case, and does not require a priori spectral knowledge. We review and extend background results on previously developed static momentum accelerations for the power iteration through the connection between the momentum accelerated iteration and the standard power iteration applied to an augmented matrix. We show that the augmented matrix is defective for the optimal parameter choice. We then present our dynamic method which updates the momentum parameter at each iteration based on the Rayleigh quotient and two previous residuals. We present convergence and stability theory for the method by considering a power‐like method consisting of multiplying an initial vector by a sequence of augmented matrices. We demonstrate the developed method on a number of benchmark problems, and see that it outperforms both the power iteration and often the static momentum acceleration with optimal parameter choice. Finally, we present and demonstrate an explicit extension of the algorithm to inverse power iterations.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"47 1","pages":""},"PeriodicalIF":4.3,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142222842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jorge Delgado, Plamen Koev, Ana Marco, José‐Javier Martínez, Juan Manuel Peña, Per‐Olof Persson, Steven Spasov
We present a new decomposition of a Cauchy–Vandermonde matrix as a product of bidiagonal matrices which, unlike its existing bidiagonal decompositions, is now valid for a matrix of any rank. The new decompositions are insusceptible to the phenomenon known as subtractive cancellation in floating point arithmetic and are thus computable to high relative accuracy. In turn, other accurate matrix computations are also possible with these matrices, such as eigenvalue computation amongst others.
{"title":"Accurate bidiagonal decompositions of Cauchy–Vandermonde matrices of any rank","authors":"Jorge Delgado, Plamen Koev, Ana Marco, José‐Javier Martínez, Juan Manuel Peña, Per‐Olof Persson, Steven Spasov","doi":"10.1002/nla.2579","DOIUrl":"https://doi.org/10.1002/nla.2579","url":null,"abstract":"We present a new decomposition of a Cauchy–Vandermonde matrix as a product of bidiagonal matrices which, unlike its existing bidiagonal decompositions, is now valid for a matrix of any rank. The new decompositions are insusceptible to the phenomenon known as subtractive cancellation in floating point arithmetic and are thus computable to high relative accuracy. In turn, other accurate matrix computations are also possible with these matrices, such as eigenvalue computation amongst others.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"55 1","pages":""},"PeriodicalIF":4.3,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141946875","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Vladislav Pimanov, Oleg Iliev, Ivan Oseledets, Ekaterina Muravleva
It is known (see, e.g., [SIAM J. Matrix Anal. Appl. 2014;35(1):143‐173]) that the performance of iterative methods for solving the Stokes problem essentially depends on the quality of the preconditioner for the Schur complement matrix, . In this paper, we consider two preconditioners for : the identity one and the SIMPLE one, and numerically study their performance for solving the Stokes problem in tight geometries. The latter are characterized by a high surface‐to‐volume ratio. We show that for such geometries, can become severely ill‐conditioned, having a very large condition number and a significant portion of non‐unit eigenvalues. As a consequence, the identity matrix, which is broadly used as a preconditioner for solving the Stokes problem in simple geometries, becomes very inefficient. We show that there is a correlation between the surface‐to‐volume ratio and the condition number of : the latter increases with the increase of the former. We show that the condition number of the diffusive SIMPLE‐preconditioned Schur complement matrix remains bounded when the surface‐to‐volume ratio increases, which explains the robust performance of this preconditioner for tight geometries. Further on, we use a direct method to calculate the full spectrum of and show that there is a correlation between the number of its non‐unit eigenvalues and the number of grid points at which no‐slip boundary conditions are prescribed. To illustrate the above findings, we examine the Pressure Schur Complement formulation for staggered finite‐difference discretization of the Stokes equations and solve it with the preconditioned conjugate gradient method. The practical problem which is of interest to us is computing the permeability of tight rocks.
{"title":"On the efficient preconditioning of the Stokes equations in tight geometries","authors":"Vladislav Pimanov, Oleg Iliev, Ivan Oseledets, Ekaterina Muravleva","doi":"10.1002/nla.2581","DOIUrl":"https://doi.org/10.1002/nla.2581","url":null,"abstract":"It is known (see, e.g., [SIAM J. Matrix Anal. Appl. 2014;35(1):143‐173]) that the performance of iterative methods for solving the Stokes problem essentially depends on the quality of the preconditioner for the Schur complement matrix, . In this paper, we consider two preconditioners for : the identity one and the SIMPLE one, and numerically study their performance for solving the Stokes problem in tight geometries. The latter are characterized by a high surface‐to‐volume ratio. We show that for such geometries, can become severely ill‐conditioned, having a very large condition number and a significant portion of non‐unit eigenvalues. As a consequence, the identity matrix, which is broadly used as a preconditioner for solving the Stokes problem in simple geometries, becomes very inefficient. We show that there is a correlation between the surface‐to‐volume ratio and the condition number of : the latter increases with the increase of the former. We show that the condition number of the diffusive SIMPLE‐preconditioned Schur complement matrix remains bounded when the surface‐to‐volume ratio increases, which explains the robust performance of this preconditioner for tight geometries. Further on, we use a direct method to calculate the full spectrum of and show that there is a correlation between the number of its non‐unit eigenvalues and the number of grid points at which no‐slip boundary conditions are prescribed. To illustrate the above findings, we examine the Pressure Schur Complement formulation for staggered finite‐difference discretization of the Stokes equations and solve it with the preconditioned conjugate gradient method. The practical problem which is of interest to us is computing the permeability of tight rocks.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"232 1","pages":""},"PeriodicalIF":4.3,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863876","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we introduce a nonconvex tensor recovery approach, which employs the powerful ket augmentation technique to expand a low order tensor into a high‐order one so that we can exploit the advantage of tensor train (TT) decomposition tailored for high‐order tensors. Moreover, we define a new nonconvex surrogate function to approximate the tensor rank, and develop an auto‐weighted mechanism to adjust the weights of the resulting high‐order tensor's TT ranks. To make our approach robust, we add two mode‐unfolding regularization terms to enhance the model for the purpose of exploring spatio‐temporal continuity and self‐similarity of the underlying tensors. Also, we propose an implementable algorithm to solve the proposed optimization model in the sense that each subproblem enjoys a closed‐form solution. A series of numerical results demonstrate that our approach works well on recovering color images and videos.
在本文中，我们介绍了一种非凸张量恢复方法，该方法利用强大的 ket 增强技术将低阶张量扩展为高阶张量，从而利用为高阶张量量身定制的张量列车（TT）分解的优势。此外，我们定义了一个新的非凸替代函数来近似张量秩，并开发了一种自动加权机制来调整由此产生的高阶张量的 TT 秩的权重。为了使我们的方法具有鲁棒性，我们添加了两个模式解折正则化项来增强模型，以探索底层张量的时空连续性和自相似性。此外，我们还提出了一种可实施的算法来解决所提出的优化模型，即每个子问题都有一个闭式解。一系列数值结果表明，我们的方法在恢复彩色图像和视频时效果良好。
{"title":"Robust tensor recovery via a nonconvex approach with ket augmentation and auto‐weighted strategy","authors":"Wenhui Xie, Chen Ling, Hongjin He, Lei‐Hong Zhang","doi":"10.1002/nla.2580","DOIUrl":"https://doi.org/10.1002/nla.2580","url":null,"abstract":"In this article, we introduce a nonconvex tensor recovery approach, which employs the powerful ket augmentation technique to expand a low order tensor into a high‐order one so that we can exploit the advantage of tensor train (TT) decomposition tailored for high‐order tensors. Moreover, we define a new nonconvex surrogate function to approximate the tensor rank, and develop an auto‐weighted mechanism to adjust the weights of the resulting high‐order tensor's TT ranks. To make our approach robust, we add two mode‐unfolding regularization terms to enhance the model for the purpose of exploring spatio‐temporal continuity and self‐similarity of the underlying tensors. Also, we propose an implementable algorithm to solve the proposed optimization model in the sense that each subproblem enjoys a closed‐form solution. A series of numerical results demonstrate that our approach works well on recovering color images and videos.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"25 1","pages":""},"PeriodicalIF":4.3,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863965","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We address the problem of finding the nearest graph Laplacian to a given matrix, with the distance measured using the Frobenius norm. Specifically, for the directed graph Laplacian, we propose two novel algorithms by reformulating the problem as convex quadratic optimization problems with a special structure: one based on the active set method and the other on direct computation of Karush–Kuhn–Tucker points. The proposed algorithms can be applied to system identification and model reduction problems involving Laplacian dynamics. We demonstrate that these algorithms possess lower time complexities and the finite termination property, unlike the interior point method and V‐FISTA, the latter of which is an accelerated projected gradient method. Our numerical experiments confirm the effectiveness of the proposed algorithms.
{"title":"The nearest graph Laplacian in Frobenius norm","authors":"Kazuhiro Sato, Masato Suzuki","doi":"10.1002/nla.2578","DOIUrl":"https://doi.org/10.1002/nla.2578","url":null,"abstract":"We address the problem of finding the nearest graph Laplacian to a given matrix, with the distance measured using the Frobenius norm. Specifically, for the directed graph Laplacian, we propose two novel algorithms by reformulating the problem as convex quadratic optimization problems with a special structure: one based on the active set method and the other on direct computation of Karush–Kuhn–Tucker points. The proposed algorithms can be applied to system identification and model reduction problems involving Laplacian dynamics. We demonstrate that these algorithms possess lower time complexities and the finite termination property, unlike the interior point method and V‐FISTA, the latter of which is an accelerated projected gradient method. Our numerical experiments confirm the effectiveness of the proposed algorithms.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"4 1","pages":""},"PeriodicalIF":4.3,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141781087","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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