Alternating Projections and Douglas-Rachford for Sparse Affine Feasibility

• Published in: IEEE transactions on signal processing Vol. 62; no. 18; pp. 4868 - 4881
• Main Authors: Hesse, Robert, Luke, D. Russell, Neumann, Patrick
• Format: Journal Article
• Language: English
• Published: New York IEEE 15.09.2014; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Abstracts; Algorithms; Compressed sensing; Convergence; Feasibility; Heuristic; iterative methods; linear systems; Mathematical analysis; Mathematical models; Optimization; Projection; Projection algorithms; Signal processing algorithms; Sparse matrices; Task analysis; Transaction processing; Vectors
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2014.2339801

The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NP-complete problem that is typically solved numerically via convex heuristics or nicely-behaved nonconvex relaxations. In this work we consider elementary methods based on projections for solving a sparse feasibility problem without employing convex heuristics. It has been shown recently that, locally, the fundamental method of alternating projections must converge linearly to a solution to the sparse feasibility problem with an affine constraint. In this paper we apply different analytical tools that allow us to show global linear convergence of alternating projections under familiar constraint qualifications. These analytical tools can also be applied to other algorithms. This is demonstrated with the prominent Douglas-Rachford algorithm where we establish local linear convergence of this method applied to the sparse affine feasibility problem.