Regularization of $\ell_1$ minimization for dealing with outliers and noise in Statistics and Signal Recovery
This work addresses robustness in statistical estimation and signal recovery, offering an incremental improvement for handling outliers and noise in linear models.
The paper tackles the problem of linear regression with outliers and noise in the dependent variable by introducing a new estimator that regularizes ℓ₁ minimization with the ℓ₂ norm, maintaining robustness to large outliers while reducing to least squares in outlier-free cases, with numerical experiments confirming theoretical properties.
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.