Alternating Projections and Douglas-Rachford for Sparse Affine Feasibility

作者：Hesse Robert^*; Luke D Russell; Neumann Patrick

来源：IEEE Transactions on Signal Processing, 2014, 62(18): 4868-4881.

DOI：10.1109/TSP.2014.2339801

• Compressed sensing
• iterative methods
• linear systems
• optimization
• projection algorithms
• sparse matrices

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 non-convex 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.

• 出版日期2014-9-15