  The recovery of sparse data is at the core of many applications in machine
learning and signal processing. While such problems can be tackled using
$\ell_1$-regularization as in the LASSO estimator and in the Basis Pursuit
approach, specialized algorithms are typically required to solve the
corresponding high-dimensional non-smooth optimization for large instances.
Iteratively Reweighted Least Squares (IRLS) is a widely used algorithm for this
purpose due its excellent numerical performance. However, while existing theory
is able to guarantee convergence of this algorithm to the minimizer, it does
not provide a global convergence rate. In this paper, we prove that a variant
of IRLS converges with a global linear rate to a sparse solution, i.e., with a
linear error decrease occurring immediately from any initialization, if the
measurements fulfill the usual null space property assumption. We support our
theory by numerical experiments showing that our linear rate captures the
correct dimension dependence. We anticipate that our theoretical findings will
lead to new insights for many other use cases of the IRLS algorithm, such as in
low-rank matrix recovery.
