Alternating Projections and Douglas-Rachford for Sparse Affine Feasibility

TL;DR

This paper demonstrates that elementary projection-based methods, specifically alternating projections and Douglas-Rachford, can globally converge linearly to solutions in sparse affine feasibility problems without convex relaxations.

Contribution

The paper proves global linear convergence of alternating projections and local linear convergence of Douglas-Rachford for sparse affine feasibility, extending previous local results.

Findings

Global linear convergence of alternating projections

Local linear convergence of Douglas-Rachford

Applicable to other algorithms with similar tools

Abstract

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. In a recent paper Bauschke, Luke, Phan and Wang (2014) showed 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.