Alternating projections on non-tangential manifolds

2011/07/08 by Andersson, Fredrik, Carlsson, Marcus · 2 citations
#FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1107.4055

Abstract

We consider sequences (B_k)_k=0^∞ of points obtained by projecting back and forth between two manifolds \M₁ and \M₂, and give conditions guaranteeing that the sequence converge to a limit B_∞∈\M₁∩\M₂. Our motivation is the study of algorithms based on finding the limit of such sequences, which have proven useful in a number of areas. The intersection is typically a set with desirable properties, but for which there is no efficient method of finding the closest point B_opt in \M₁∩\M₂. We prove not only that the sequence of alternating projections converges, but that the limit point is fairly close to B_opt, in a manner relative to the distance ‖B₀-B_opt‖, thereby significantly improving earlier results in the field. A concrete example with applications to frequency estimation of signals is also presented.

Cited by

• Optimal structured approximation of Fourier subspaces, Toeplitz matrices, and exponential sums
• Regularization in Data-driven Predictive Control: A Convex Relaxation Perspective

Related

• Fast Alternating Projections on Manifolds Based on Tangent Spaces
• Inexact alternating projections on nonconvex sets
• Local convergence for alternating and averaged nonconvex projections
• On local convergence of the method of alternating projections
• Quasioptimal alternating projections and their use in low-rank approximation of matrices and tensors