Projection-like Retractions on Matrix Manifolds

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.