Paper ID sheet UCL-INMA-2010.038


Projection-like retractions on matrix manifolds

P.-A. Absil, Jérôme Malick
This paper deals with contructing retractions, a key step when applying optimization algorithms on matrix manifolds. For submanifolds of Euclidean spaces, we show that the operation consisting of taking a tangent step in the embedding Euclidean space followed by a projection onto the submanifold, is a retraction. We also show that the operation remains a retraction if the projection is generalized to a projection-like procedure that consists of coming back to the submanifold along ``admissible'' directions, and we give a sufficient condition on the admissible directions for the generated retraction to be second order. This theory offers a framework in which previously-proposed retractions can be analyzed, as well as a toolbox for constructing new ones. Illustrations are given for projections-like procedures on some specific manifolds for which we have an explicit, easy-to-compute expression.
Key words
equality-constrained optimization; matrix manifold; feasible optimization method; retraction; projection; fixed-rank matrices; Stiefel manifold; spectral manifold
SIAM Journal on Optimization, Volume 22, Issue 1, pp. 135-158, 2012
BibTeX entry

author = {P.-A. Absil and J\'{e}r\^{o}me Malick},
title = {Projection-like Retractions on Matrix Manifolds},
publisher = {SIAM},
year = {2012},
journal = {SIAM Journal on Optimization},
volume = {22},
number = {1},
pages = {135-158},
url = {},
doi = {10.1137/100802529}
The errata concerns the publisher's version, doi:10.1137/100802529.