Low-rank retractions: a survey and new results

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 M r of rank- r m × n matrices. With the exception of the exponential retraction...

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Bibliographic Details
Published inComputational optimization and applications Vol. 62; no. 1; pp. 5 - 29
Main Authors Absil, P.-A., Oseledets, I. V.
Format Journal Article
LanguageEnglish
Published New York Springer US 01.09.2015
Springer Nature B.V
Subjects
Analysis
Computation
Convex and Discrete Geometry
Counting
Euclidean space
Image processing systems
Management Science
Manifolds (mathematics)
Mathematical functions
Mathematics
Mathematics and Statistics
Operations Research
Operations Research/Decision Theory
Optimization
Run time (computers)
Statistics
Studies
Symmetry
Texts
Topological manifolds
Orthographic retraction
Low-rank optimization
Retraction
Projective retraction
Low-rank manifold
Geodesic
Fixed-rank manifold
Lie–Trotter splitting
Quasi-geodesic
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Summary: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 M r of rank- r m × 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.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
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ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-014-9714-4