  In this paper we theoretically study exact recovery of sparse vectors from
compressed measurements by minimizing a general nonconvex function that can be
decomposed into the sum of single variable functions belonging to a class of
smooth nonconvex sparsity promoting functions. Null space property (NSP) and
restricted isometry property (RIP) are used as key theoretical tools. The
notion of \emph{scale function} associated to a sparsity promoting function is
introduced to generalize the state-of-the-art analysis technique of the $l_p$
minimization problem. The analysis is used to derive an upper bound on the null
space constant (NSC) associated to this general nonconvex minimization problem,
which is further utilized to derive sufficient conditions for exact recovery as
upper bounds on the restricted isometry constant (RIC), as well as bounds on
optimal sparsity $K$ for which exact recovery occurs. The derived bounds are
explicitly calculated when the sparsity promoting function $f$ under
consideration possesses the property that the associated \emph{elasticity
function}, defined as, $\psi(x)=\frac{xdf(x)/dx}{f(x)}$, is monotonic in
nature. Numerical simulations are carried out to verify the efficacy of the
bounds and interesting conclusions are drawn about the comparative performances
of different sparsity promoting functions for the problem of $1$-sparse signal
recovery.
