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Summary: Technical report UCL-INMA-2010.038
Projection-like retractions on matrix manifolds
P.-A. Absil
J´er^ome Malick
July 16, 2010
Abstract
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.
Key words. Equality-constrained optimization, matrix manifold, feasible optimization
method, retraction, projection, fixed-rank matrices, Stiefel manifold, spectral manifold
1 Introduction
Computational problems abound that can be formulated as finding an optimal point of a
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