  Compressed sensing is a central topic in signal processing with myriad
applications, where the goal is to recover a signal from as few observations as
possible. Iterative re-weighting is one of the fundamental tools to achieve
this goal. This paper re-examines the iteratively reweighted least squares
(IRLS) algorithm for sparse recovery proposed by Daubechies, Devore, Fornasier,
and G\"unt\"urk in \emph{Iteratively reweighted least squares minimization for
sparse recovery}, {\sf Communications on Pure and Applied Mathematics}, {\bf
63}(2010) 1--38. Under the null space property of order $K$, the authors show
that their algorithm converges to the unique $k$-sparse solution for $k$
strictly bounded above by a value strictly less than $K$, and this $k$-sparse
solution coincides with the unique $\ell_1$ solution. On the other hand, it is
known that, for $k$ less than or equal to $K$, the $k$-sparse and $\ell_1$
solutions are unique and coincide. The authors emphasize that their proof
method does not apply for $k$ sufficiently close to $K$, and remark that they
were unsuccessful in finding an example where the algorithm fails for these
values of $k$.
  In this note we construct a family of examples where the
Daubechies-Devore-Fornasier-G\"unt\"urk IRLS algorithm fails for $k=K$, and
provide a modification to their algorithm that provably converges to the unique
$k$-sparse solution for $k$ less than or equal to $K$ while preserving the
local linear rate. The paper includes numerical studies of this family as well
as the modified IRLS algorithm, testing their robustness under perturbations
and to parameter selection.
