Exact recoverability from dense corrupted observations via $L_1$ minimization

This paper presents a surprising phenomenon: given $m$ highly corrupted measurements $y = A_{\Omega \bullet} x^{\star} + e^{\star}$, where $A_{\Omega \bullet}$ is a submatrix selected uniformly at random from 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 $l_1$-minimization can recover $x^{\star}$ 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 $100$ 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 dense noise with bounded energy. Simulation results under various settings are also presented to verify the validity of the theory as well as to illustrate the promising potentials of the proposed framework.