Robust penalized dantzig selector: Error analysis, oracle inequalities, and algorithmic efficiency

This article addresses the recovery of sparse signals and low-rank matrices from limited, noisy observations. We introduce unconstrained models, termed the Robust Penalized Dantzig Vector Selector (RPDvS) for sparse signal recovery and the Robust Penalized Dantzig Matrix Selector (RPDmS) for low-rank matrix recovery. Under restricted isometry assumptions on the measurement operators, we establish stable recovery error bounds and oracle inequalities for both models in the presence of Gaussian noise. For the RPDmS model, we derive a tight oracle inequality that guarantees identification of a 'nearly perfect'' column space and motivates a simple post-processing procedure that yields an improved estimator, thereby bridging a gap in the earlier analysis of the Dantzig matrix selector by Candès and Plan (2011). Parallel oracle-type results are obtained for the RPDvS model.Building on these theoretical guarantees, we develop efficient algorithms for solving the RPDvS and RPDmS problems and implement the proposed post-processing steps. Extensive simulations and empirical studies demonstrate that the new methods are computationally competitive while achieving superior reconstruction accuracy in both sparse signal and low-rank matrix recovery, thereby confirming the sharpness of our error bounds and the effectiveness of the algorithms.