  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.
