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In this paper, we analyse the recovery properties of nonconvex regularized$M$-estimators, under the assumption that the true parameter is of softsparsity. In the statistical aspect, we establish the recovery bound for anystationary point of the nonconvex regularized $M$-estimator, under restrictedstrong convexity and some regularity conditions on the loss function and theregularizer, respectively. In the algorithmic aspect, we slightly decompose theobjective function and then solve the nonconvex optimization problem via theproximal gradient method, which is proved to achieve a linear convergence rate.In particular, we note that for commonly-used regularizers such as SCAD andMCP, a simpler decomposition is applicable thanks to our assumption on theregularizer, which helps to construct the estimator with better recoveryperformance. Finally, we demonstrate our theoretical consequences and theadvantage of the assumption by several numerical experiments on the corruptederrors-in-variables linear regression model. Simulation results show remarkableconsistency with our theory under high-dimensional scaling.

Sparse recovery via nonconvex regularized $M$-estimators over $\ell_q$-balls