  Finding the sparse solution of an underdetermined system of linear equations
has many applications, especially, it is used in Compressed Sensing (CS),
Sparse Component Analysis (SCA), and sparse decomposition of signals on
overcomplete dictionaries. We have recently proposed a fast algorithm, called
Smoothed $\ell^0$ (SL0), for this task. Contrary to many other sparse recovery
algorithms, SL0 is not based on minimizing the $\ell^1$ norm, but it tries to
directly minimize the $\ell^0$ norm of the solution. The basic idea of SL0 is
optimizing a sequence of certain (continuous) cost functions approximating the
$\ell^0$ norm of a vector. However, in previous papers, we did not provide a
complete convergence proof for SL0. In this paper, we study the convergence
properties of SL0, and show that under a certain sparsity constraint in terms
of Asymmetric Restricted Isometry Property (ARIP), and with a certain choice of
parameters, the convergence of SL0 to the sparsest solution is guaranteed.
Moreover, we study the complexity of SL0, and we show that whenever the
dimension of the dictionary grows, the complexity of SL0 increases with the
same order as Matching Pursuit (MP), which is one of the fastest existing
sparse recovery methods, while contrary to MP, its convergence to the sparsest
solution is guaranteed under certain conditions which are satisfied through the
choice of parameters.
