Low-rank retractions: a survey and new results

• Published in: Computational optimization and applications Vol. 62; no. 1; pp. 5 - 29
• Main Authors: Absil, P.-A., Oseledets, I. V.
• Format: Journal Article
• Language: English
• 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

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...