Projection-like Retractions on Matrix Manifolds

• Published in: SIAM journal on optimization Vol. 22; no. 1; pp. 135 - 158
• Main Authors: Absil, P.-A., Malick, Jérôme
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2012
• Subjects: Approximation; Euclidean space; Lie groups; Mathematical functions; Mathematics; Methods; Optimization algorithms; Optimization and Control; retraction; Stiefel manifold; spectral manifold; feasible optimization method; projection; matrix manifold; fixed-rank matrices; equality-constrained optimization
• ISSN: 1052-6234; 1095-7189
• DOI: 10.1137/100802529

This paper deals with constructing 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 projection-like procedures on some specific manifolds for which we have an explicit, easy-to-compute expression.