## Intrinsic Representation of Tangent Vectors and Vector transport on Matrix Manifolds

### Authors

Wen Huang, P.-A. Absil, K. A. Gallivan
### Abstract

In Riemannian optimization problems, commonly encountered manifolds are $d$-dimensional matrix manifolds whose tangent spaces can be represented by $d$-dimensional linear subspaces of a $w$-dimensional Euclidean space, where $w > d$. Therefore, representing tangent vectors by $w$-dimensional vectors has been commonly used in practice. However, using $w$-dimensional vectors may be the most natural but may not be the most efficient approach. A recent paper, [Mathematical Programming Series A, 150:2, pp. 179-216, 2014], proposed using $d$-dimensional vectors to represent tangent vectors and showed its benefits without giving detailed implementations for commonly encountered manifolds. In this paper, we discuss the implementations of using $d$-dimensional vectors to represent tangent vectors for the Stiefel manifold, the Grassmann manifold, the fixed-rank manifold and the manifold of positive semidefinite matrices with rank fixed. A Riemannian quasi-Newton method for minimizing the Brockett cost function is used to demonstrate the performance of the $d$-dimensional representation.
### Status

Accepted in *Numerische Mathematik*, DOI:10.1007/s00211-016-0848-4, 2016.
### Download

- Technical report: PDF
- Experiment code: Code

### BibTex entry

- Technical Report

@TECHREPORT{HAG2016,

author = "Wen Huang and P.-A. Absil and K. A. Gallivan",

title = "Intrinsic Representation of Tangent Vectors and Vector transport on Matrix Manifolds",

institution = "U.C.Louvain",

number = "UCL-INMA-2016.08.v2",

year = 2016,

}

- Published version

@article{HAG2016,

author = "Wen Huang and P.-A. Absil and K. A. Gallivan",

title = "Intrinsic Representation of Tangent Vectors and Vector transport on Matrix Manifolds",

journal = "Numerische Mathematik",

doi = "10.1007/s00211-016-0848-4",

year = 2016,

}