标题： 通过非凸正则化$M$-估计器在$\ell_q$-球上的稀疏恢复 显示英文标题

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

摘要： 在本文中，我们在真实参数具有软稀疏性的假设下，分析了非凸正则化$M$估计量的恢复性质。在统计方面，我们在损失函数和正则化项分别满足限制强凸性和某些正则性条件下，建立了非凸正则化$M$估计量任何平稳点的恢复界。在算法方面，我们对目标函数进行轻微分解，然后通过近端梯度方法解决非凸优化问题，该方法被证明具有线性收敛速率。特别地，我们注意到对于常用的正则化方法如 SCAD 和 MCP，在我们对正则化项的假设下，可以应用更简单的分解方式，这有助于构建具有更好恢复性能的估计量。最后，我们通过在损坏误差变量线性回归模型上的几个数值实验，展示了我们的理论结果以及该假设的优势。模拟结果在高维尺度下与我们的理论表现出显著的一致性。 显示英文摘要

摘要： 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.