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In this paper we theoretically study exact recovery of sparse vectors fromcompressed measurements by minimizing a general nonconvex function that can bedecomposed into the sum of single variable functions belonging to a class ofsmooth nonconvex sparsity promoting functions. Null space property (NSP) andrestricted isometry property (RIP) are used as key theoretical tools. Thenotion of \emph{scale function} associated to a sparsity promoting function isintroduced 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 nullspace constant (NSC) associated to this general nonconvex minimization problem,which is further utilized to derive sufficient conditions for exact recovery asupper bounds on the restricted isometry constant (RIC), as well as bounds onoptimal sparsity $K$ for which exact recovery occurs. The derived bounds areexplicitly calculated when the sparsity promoting function $f$ underconsideration possesses the property that the associated \emph{elasticityfunction}, defined as, $\psi(x)=\frac{xdf(x)/dx}{f(x)}$, is monotonic innature. Numerical simulations are carried out to verify the efficacy of thebounds and interesting conclusions are drawn about the comparative performancesof different sparsity promoting functions for the problem of $1$-sparse signalrecovery.

Sparse Recovery Analysis of Generalized $J$-Minimization with Results for Sparsity Promoting Functions with Monotonic Elasticity