A Riemannian Optimization Approach for Solving the Generalized Eigenvalue Problem for Nonsquare Matrix Pencils

Li, Jiao‐fen; Li, Wen; Vong, Seakweng; Luo, Qilun; Xiao, Mingqing

doi:10.1007/s10915-020-01173-5

Cited by 13 publications

(15 citation statements)

References 18 publications

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“…and where M v andP are defined as in (19). Hence, by using the projection P (V, P ) given in (10), we obtain grad f (V, P) = P (V,P) ( ∇f (V, P)…”

Section: Riemannian Gradient and Hessianmentioning

confidence: 99%

“…The Riemannian Hessian Hess f of f on St(n, l, C) × St(2l, l, C) can also be expressed by using ∇ 2 f and the projection map (10). That is,…”

Section: Riemannian Gradient and Hessianmentioning

confidence: 99%

“…Recently, we reformulate the problem (4) as an optimization problem on the product of two complex Stiefel manifolds and then develop an algorithm based on the Riemannian nonlinear conjugate gradient method. 10 Although the algorithm is shown to be effective, the Riemannian nonlinear-conjugate-gradient-based method does not provide an ideal efficiency, in particular, for the pencils with large sizes. This motivates us to develop a more efficient algorithm such as some second-order approaches for solving the underlying problem.…”

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confidence: 99%

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A trust‐region method for the parameterized generalized eigenvalue problem with nonsquare matrix pencils

Wang

Liu

et al. 2021

Numerical Linear Algebra App

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The l parameterized generalized eigenvalue problems for the nonsquare matrix pencils, proposed by Chu and Golub [SIAM J. Matrix Anal. Appl., 28(2006), pp. 770-787], can be formulated as an optimization problem on a corresponding complex product Stiefel manifold. Some early proposed algorithms are based on the first-order information of the objective function, and fast convergence could not be expected. In this article, we turn the generic Riemannian trust-region method of Absil et al. into a practical algorithm for solving the underlying problem, which enjoys the global convergence and local superlinear convergence rate. Numerical experiments are provided to illustrate the efficiency of the proposed method. Detailed comparisons with some existing results (l = 1 and l = n) are given first. For the case l = n, the algorithm can yield exactly the same optimal solution as Ito and Murota's algorithm, which is essentially a direct method to solve the problem. The algorithm runs faster than Boutry et al.'s algorithm for the case l = 1. Further comparisons with some early proposed gradient-based algorithms, and some latest infeasible methods for solving manifold optimization problems are also provided to show the merits of the proposed approach.

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“…and where M v andP are defined as in (19). Hence, by using the projection P (V, P ) given in (10), we obtain grad f (V, P) = P (V,P) ( ∇f (V, P)…”

Section: Riemannian Gradient and Hessianmentioning

confidence: 99%

“…The Riemannian Hessian Hess f of f on St(n, l, C) × St(2l, l, C) can also be expressed by using ∇ 2 f and the projection map (10). That is,…”

Section: Riemannian Gradient and Hessianmentioning

confidence: 99%

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confidence: 99%

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A trust‐region method for the parameterized generalized eigenvalue problem with nonsquare matrix pencils

Wang

Liu

et al. 2021

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“…By using the classical expression of the Riemannian connection on a Riemannian submanifold of a Euclidean space (see [32], §5.3.3) and choosing ∇ ξ ζ ≔ P X (Dζ (X)[ξ]), the Riemannian Hessian hess f of f on St(n, p) can also be expressed by using ∇ 2 f and the projection map (15). at is,…”

Section: Lemmamentioning

confidence: 99%

“…Optimization over the Stiefel manifold is an important special case of Riemannian optimization, which has recently aroused considerable research interests due to the wide applications in different fields such as the linear eigenvalue problem, the orthogonal Procrustes problem, the nearest low-rank correlation matrix problem, the Kohn-Sham total energy minimization, and singular value decomposition. Since optimization over the Stiefel manifold can be viewed as a general nonlinear optimization problem with constraints, many standard algorithms [11] in the Euclidean space can be generalized to manifold setting directly and have been explored and successfully applied to various applications, e.g., Riemannian steepest descent method [12], Riemannian curvilinear search method with Barzilai-Borwein (BB) steps [13], Riemannian Dai's nonmonotone-based conjugate gradient method [14,15], Riemannian Polak-Ribière-Polyak-based nonlinear conjugate gradient method [16,17], and Riemannian Fletcher-Reeves-based conjugate gradient method [18]. However, as we know, gradient-type algorithms often perform reasonably well but might converge slowly when the generated iterates are close to an optimal solution.…”

Section: Introductionmentioning

confidence: 99%

Effective Algorithms for Solving Trace Minimization Problem in Multivariate Statistics

Wen

Zhou

et al. 2020

Mathematical Problems in Engineering

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This paper develops two novel and fast Riemannian second-order approaches for solving a class of matrix trace minimization problems with orthogonality constraints, which is widely applied in multivariate statistical analysis. The existing majorization method is guaranteed to converge but its convergence rate is at best linear. A hybrid Riemannian Newton-type algorithm with both global and quadratic convergence is proposed firstly. A Riemannian trust-region method based on the proposed Newton method is further provided. Some numerical tests and application to the least squares fitting of the DEDICOM model and the orthonormal INDSCAL model are given to demonstrate the efficiency of the proposed methods. Comparisons with some latest Riemannian gradient-type methods and some existing Riemannian second-order algorithms in the MATLAB toolbox Manopt are also presented.

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Gradient based iterative methods for solving symmetric tensor equations

Wang

Zhang

2021

Numerical Linear Algebra App

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The steepest descent and conjugate gradient methods are classical gradient based iterative methods for solving symmetric positive definite linear system Ax=b. In this article, we are concerned with the numerical solution of the tensor equation 𝒜xm−1=b by those well‐known iterations in which 𝒜 is an mth order n‐dimensional symmetric tensor. Then we prove that the developed iterative methods converge locally linearly under some appropriate conditions. Finally, some numerical experiments are provided to illustrate the feasibility and effectiveness of the proposed methods.

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A Riemannian Optimization Approach for Solving the Generalized Eigenvalue Problem for Nonsquare Matrix Pencils

Cited by 13 publications

References 18 publications

A trust‐region method for the parameterized generalized eigenvalue problem with nonsquare matrix pencils

A trust‐region method for the parameterized generalized eigenvalue problem with nonsquare matrix pencils

Effective Algorithms for Solving Trace Minimization Problem in Multivariate Statistics

Gradient based iterative methods for solving symmetric tensor equations

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