Hyperparameter-free Weighted Median Operator for Robust Spectral Analysis in Heavy-Tailed Noise Environment

The least absolute shrinkage and selection operator (LASSO) regression has been widely used for high-resolution spectral line estimation. Since the derivation of the LASSO is based on
$l$
-2 norm, it loses robustness in environments with heavy-tailed noise. combining the LAD and LASSO criteria, the least absolution deviation (LAD)-LASSO can effectively resist outliers and heavy-tailed errors. However, this method encounters the problem of optimal artificial parameter selection. In this paper, a robust Semi-Parametric/SParse Iterative Covariance-based Estimator (SPICE) is first introduced, which significantly eases parameter selection. Secondly, a weighted median algorithm is uesd to robustly estimate each frequency coefficient in a coordinate descent framework. The proposed method can effectively resist outliers and heavy-tailed errors without tuning user parameter, and greatly improve the computational efficiency compared with traditional SPICE. Simulations verify the effectiveness of the proposed algorithm.