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Intrinsic Representation of Tangent Vectors and Vector transport on Matrix Manifolds

Wen Huang(huwst08***at***gmail.com)
P.-A. Absil(absil***at***inma.ucl.ac.be)
Kyle Gallivan(kgallivan***at***fsu.edu)

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.

Keywords: Vector transport; Riemannian quasi-Newton methods; Intrinsic representation; Matrix manifolds

Category 1: Nonlinear Optimization

Category 2: Applications -- Science and Engineering

Category 3: Linear, Cone and Semidefinite Programming

Citation: Universite catholique de Louvain; Tech. report-UCL-INMA-2016.08; May 2016

Download: [PDF]

Entry Submitted: 05/18/2016
Entry Accepted: 05/18/2016
Entry Last Modified: 05/18/2016

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