Cholesky QR-based retraction on the generalized Stiefel manifold

被引：0
作者：
Hiroyuki Sato
Kensuke Aihara
机构：
[1] Kyoto University,The Hakubi Center for Advanced Research
[2] Kyoto University,Department of Applied Mathematics and Physics
[3] Tokyo City University,Department of Computer Science
来源：
Computational Optimization and Applications | 2019年 / 72卷
关键词：
Riemannian optimization; Generalized Stiefel manifold; Retraction; Cholesky QR factorization; 90C30; 65K05; 65F30;
D O I：
暂无
中图分类号：
学科分类号：
摘要：
When optimizing on a Riemannian manifold, it is important to use an efficient retraction, which maps a point on a tangent space to a point on the manifold. In this paper, we prove a map based on the QR factorization to be a retraction on the generalized Stiefel manifold. In addition, we propose an efficient implementation of the retraction based on the Cholesky QR factorization. Numerical experiments show that the proposed retraction is more efficient than the existing one based on the polar factorization.
引用
收藏
页码：293 / 308
页数：15
相关论文
共 33 条
[1]  
Absil PA(2012)Projection-like retractions on matrix manifolds SIAM J. Optim. 22 135-158
[2]  
Malick J(2002)Newton’s method on Riemannian manifolds and a geometric model for the human spine IMA J. Numer. Anal. 22 359-390
[3]  
Adler RL(2017)A matrix-free implementation of Riemannian Newton’s method on the Stiefel manifold Optim. Lett. 11 1729-1741
[4]  
Dedieu JP(2018)Non-convex phase retrieval from STFT measurements IEEE Trans. Inf. Theory 64 467-484
[5]  
Margulies JY(2014)Manopt, a Matlab toolbox for optimization on manifolds J. Mach. Learn. Res. 15 1455-1459
[6]  
Martens M(1998)The geometry of algorithms with orthogonality constraints SIAM J. Matrix Anal. Appl. 20 303-353
[7]  
Shub M(2002)Optimization algorithms exploiting unitary constraints IEEE Trans. Signal Process. 50 635-650
[8]  
Aihara K(2016)Riemannian preconditioning SIAM J. Optim. 26 635-660
[9]  
Sato H(2012)Optimization methods on Riemannian manifolds and their application to shape space SIAM J. Optim. 22 596-627
[10]  
Bendory T(2012)Numerical stability of orthogonalization methods with a non-standard inner product BIT Numer. Math. 52 1035-1058
1234