  In this paper, we propose a way to combine two acceleration techniques for
the $\ell\_{1}$-regularized least squares problem: safe screening tests, which
allow to eliminate useless dictionary atoms; and the use of fast structured
approximations of the dictionary matrix. To do so, we introduce a new family of
screening tests, termed stable screening, which can cope with approximation
errors on the dictionary atoms while keeping the safety of the test (i.e. zero
risk of rejecting atoms belonging to the solution support). Some of the main
existing screening tests are extended to this new framework. The proposed
algorithm consists in using a coarser (but faster) approximation of the
dictionary at the initial iterations and then switching to better
approximations until eventually adopting the original dictionary. A systematic
switching criterion based on the duality gap saturation and the screening ratio
is derived.Simulation results show significant reductions in both computational
complexity and execution times for a wide range of tested scenarios.
