In this paper, we analyse the recovery properties of nonconvex regularized $M$-estimators, under the assumption that the true parameter is of soft sparsity. In the statistical aspect, we establish the recovery bound for any stationary point of the nonconvex regularized $M$-estimator, under restricted strong convexity and some regularity conditions on the loss function and the regularizer, respectively. In the algorithmic aspect, we slightly decompose the objective function and then solve the nonconvex optimization problem via the proximal gradient method, which is proved to achieve a linear convergence rate. In particular, we note that for commonly-used regularizers such as SCAD and MCP, a simpler decomposition is applicable thanks to our assumption on the regularizer, which helps to construct the estimator with better recovery performance. Finally, we demonstrate our theoretical consequences and the advantage of the assumption by several numerical experiments on the corrupted errors-in-variables linear regression model. Simulation results show remarkable consistency with our theory under high-dimensional scaling.

中文翻译：

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通过 ℓq 球上的非凸正则化 M 估计器进行稀疏恢复

在本文中，我们在真实参数是软稀疏性的假设下分析了非凸正则化 $M$-estimators 的恢复特性。在统计方面，我们分别在限制强凸性和损失函数和正则化器的一些正则条件下，为非凸正则化$M$-estimator 的任何驻点建立恢复界限。在算法方面，我们稍微分解了目标函数，然后通过近端梯度法求解非凸优化问题，证明可以实现线性收敛速度。特别是，我们注意到对于常用的正则化器，如 SCAD 和 MCP，由于我们对正则化器的假设，更简单的分解是适用的，这有助于构建具有更好恢复性能的估计器。最后，我们通过对损坏的变量误差线性回归模型的几个数值实验证明了我们的理论结果和假设的优势。仿真结果表明在高维缩放下与我们的理论具有显着的一致性。