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 of rank- 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.