Sparse algorithms and bounds for statistically and computationally efficient robust estimation

Robust estimators that provide accurate parameter estimates even under the condition that classical assumptions like outlier-free additive Gaussian measurement noise do not hold exactly are of great practical importance in signal processing and measurement science in general. Lots of methods for deriving robust estimators exist. In this paper, we derive novel algorithms for robust estimation by modeling the outliers as a sparse additive vector of unknown deterministic or random parameters. By exploiting the separability of the estimation problem and applying recently developed sparse estimation techniques, algorithms that remove the effect of the outlying observations can be developed. Monte Carlo simulations show that the performance of the developed algorithms is practically equal to the best possible performance given by the Crámer-Rao lower bound (CRB) and the mean-squared error (MSE) of the oracle estimator [1], demonstrating the high accuracy. It is shown that the algorithms can be implemented in a computationally efficient manner. Furthermore, some interesting connections to the popular least absolute deviation (LAD) estimator are shown.