Mathematics of Operations Research
A Riemannian Optimization Approach for Computing Low-Rank Solutions of Lyapunov Equations
SIAM Journal on Matrix Analysis and Applications
Projection-like Retractions on Matrix Manifolds
SIAM Journal on Optimization
Shifted subspaces tracking on sparse outlier for motion segmentation
IJCAI'13 Proceedings of the Twenty-Third international joint conference on Artificial Intelligence
Prox-Regularity of Rank Constraint Sets and Implications for Algorithms
Journal of Mathematical Imaging and Vision
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We prove that if two smooth manifolds intersect transversally, then the method of alternating projections converges locally at a linear rate. We bound the speed of convergence in terms of the angle between the manifolds, which in turn we relate to the modulus of metric regularity for the intersection problem, a natural measure of conditioning. We discuss a variety of problem classes where the projections are computationally tractable, and we illustrate the method numerically on a problem of finding a low-rank solution of a matrix equation.