Journal
SIAM JOURNAL ON OPTIMIZATION
Volume 22, Issue 1, Pages 135-158Publisher
SIAM PUBLICATIONS
DOI: 10.1137/100802529
Keywords
equality-constrained optimization; matrix manifold; feasible optimization method; retraction; projection; fixed-rank matrices; Stiefel manifold; spectral manifold
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Funding
- Interuniversity Attraction Poles Programme
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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.
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