Computational Optimization and Applications

, Volume 72, Issue 2, pp 293–308 | Cite as

Cholesky QR-based retraction on the generalized Stiefel manifold

  • Hiroyuki SatoEmail author
  • Kensuke Aihara


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.


Riemannian optimization Generalized Stiefel manifold Retraction Cholesky QR factorization 

Mathematics Subject Classification

90C30 65K05 65F30 



The authors would like to thank the editor and the reviewer for their careful reading and constructive comments, especially on Theorem 3.2. The authors would also like to thank Dr. Akira Imakura (University of Tsukuba) and Dr. Yusaku Yamamoto (The University of Electro-Communications) for their helpful advice. This study was supported in part by Grant Numbers JP16K17647 and JP18K18064 from the Grants-in-Aid for Scientific Research Program (KAKENHI) of the Japan Society for the Promotion of Science (JSPS).


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.The Hakubi Center for Advanced ResearchKyoto UniversityKyotoJapan
  2. 2.Department of Applied Mathematics and PhysicsKyoto UniversityKyotoJapan
  3. 3.Department of Computer ScienceTokyo City UniversityTokyoJapan

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