Paper ID sheet UCL-INMA-2013.04

Title

Low-rank retractions: a survey and new results

Authors

P.-A. Absil, I. V. Oseledets

Abstract

Retractions are a prevalent tool in Riemannian optimization that provides a way to smoothly select a curve on a manifold with given initial position and velocity. We review and propose several retractions on the manifold $\mathcal{M}_r$ of rank-$r$ $m\times n$ matrices. With the exception of the exponential retraction (for the embedded geometry), which is clearly the least efficient choice, the retractions considered do not differ much in terms of run time and flop count. However, considerable differences are observed according to properties such as domain of definition, boundedness, first/second-order property, and symmetry.

Key words

Low-rank manifold; fixed-rank manifold; low-rank optimization; retraction; geodesic; quasi-geodesic; projective retraction; orthographic retraction; Lie-Trotter splitting

Status

Computational Optimization and Applications, September 2015, Volume 62, Issue 1, pp 5-29

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BibTeX entry

    @ARTICLE{AbsOse2013.04,
     author = "P.-A. Absil and I. V. Oseledets",
     title = "Low-rank retractions: a survey and new results",
     journal = "Computational Optimization and Applications",
     year = 2014,
     note = "accepted for publication",
     url = "http://sites.uclouvain.be/absil/2013.04",
    }