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Article Dans Une Revue SIAM Journal on Optimization Année : 2012

Projection-like retractions on matrix manifolds

Résumé

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 pro- jection 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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Dates et versions

hal-00651608 , version 1 (13-12-2011)
hal-00651608 , version 2 (14-12-2011)

Identifiants

Citer

Pierre-Antoine Absil, Jérôme Malick. Projection-like retractions on matrix manifolds. SIAM Journal on Optimization, 2012, 22 (1), pp.135-158. ⟨10.1137/100802529⟩. ⟨hal-00651608v2⟩
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