Matrix analysis
The Riccati equation
Geometric optimization methods for adaptive filtering
Geometric optimization methods for adaptive filtering
The Geometry of Algorithms with Orthogonality Constraints
SIAM Journal on Matrix Analysis and Applications
The $\U$-Lagrangian of the Maximum Eigenvalue Function
SIAM Journal on Optimization
Mathematical Programming: Series A and B
Trust-Region Methods on Riemannian Manifolds
Foundations of Computational Mathematics
Alternating Projections on Manifolds
Mathematics of Operations Research
Optimization Algorithms on Matrix Manifolds
Optimization Algorithms on Matrix Manifolds
Optimization algorithms exploiting unitary constraints
IEEE Transactions on Signal Processing
A Riemannian subgradient algorithm for economic dispatch with valve-point effect
Journal of Computational and Applied Mathematics
Geometric multiscale decompositions of dynamic low-rank matrices
Computer Aided Geometric Design
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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.