Exact recoverability from dense corrupted observations via $L_1$ minimization

Abstract

This paper confirms a surprising phenomenon first observed by Wright \textit{et al.} \cite{WYGSM_Face_2009_J} \cite{WM_denseError_2010_J} under different setting: given $m$ highly corrupted measurements $y = A_{\Omega \bullet} x^{\star} + e^{\star}$, where $A_{\Omega \bullet}$ is a submatrix whose rows are selected uniformly at random from rows of an orthogonal matrix $A$ and $e^{\star}$ is an unknown sparse error vector whose nonzero entries may be unbounded, we show that with high probability $\ell_1$-minimization can recover the sparse signal of interest $x^{\star}$ exactly from only $m = C \mu^2 k (\log n)^2$ where $k$ is the number of nonzero components of $x^{\star}$ and $\mu = n \max_{ij} A_{ij}^2$, even if nearly 100% of the measurements are corrupted. We further guarantee that stable recovery is possible when measurements are polluted by both gross sparse and small dense errors: $y = A_{\Omega \bullet} x^{\star} + e^{\star}+ \nu$ where $\nu$ is the small dense noise with bounded energy. Numerous simulation results under various settings are also presented to verify the validity of the theory as well as to illustrate the promising potential of the proposed framework.