Regularization of $\ell_1$ minimization for dealing with outliers and noise in Statistics and Signal Recovery

Salvador Flores; Luis M. Briceno-Arias

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arXiv · 1310.7637

Regularization of $\ell_1$ minimization for dealing with outliers and noise in Statistics and Signal Recovery

Salvador Flores·Luis M. Briceno-Arias

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Abstract

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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Salvador Flores, Luis M. Briceno-Arias. 2014-02-25. Regularization of $\ell_1$ minimization for dealing with outliers and noise in Statistics and Signal Recovery. https://arxiv.org/abs/1310.7637

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