$\ell_1$最小化的正则化处理统计与信号恢复中的野值和噪声
Regularization of $\ell_1$ minimization for dealing with outliers and noise in Statistics and Signal Recovery
摘要：
本文研究了在给定设计矩阵和污染仅限于因变量的经典线性回归问题中，$\ell_1$范数最小化的鲁棒性。我们进行了精细的误差分析的$\ell_1$估计的测量误差与噪声相结合的离群值。我们介绍了一种新的估计技术，从正规化的$\ell_1$最小化的下卷积与$\ell_2$规范。考虑到对大离群值的鲁棒性，所提出的估计保持了$\ell_1$估计的崩溃点，并在没有离群值时简化为最小二乘。我们提出了一个全局收敛的向前向后算法计算我们的估计和一些数值实验证实其理论性质。
英文摘要：
We study the robustness properties of $\ell_1$ norm minimization for the classical linear regression problem with a given design matrix and contamination restricted to the dependent variable. We perform a fine error analysis of the $\ell_1$ estimator for measurements errors consisting of outliers coupled with noise. We introduce a new estimation technique resulting from a regularization of $\ell_1$ minimization by inf-convolution with the $\ell_2$ norm. Concerning robustness to large outliers, the proposed estimator keeps the breakdown point of the $\ell_1$ estimator, and reduces to least squares when there are not outliers. We present a globally convergent forward-backward algorithm for computing our estimator and some numerical experiments confirming its theoretical properties.
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