Dense error correction via l1-minimization

We study the problem of recovering a non-negative sparse signal x isin Ropfⁿ from highly corrupted linear measurements y = Ax+e isin Ropf^m, where e is an unknown (and unbounded) error. Motivated by an observation from computer vision, we prove that for highly correlated dictionaries A, any non-negative, sufficiently sparse signal x can be recovered by solving an lscr¹-minimization problem: min ||x||₁ + ||e||₁ subject to y = Ax + e. If the fraction rho of errors is bounded away from one and the support of x grows sublinearly in the dimension m of the observation, for large m, the above lscr¹-minimization recovers all sparse signals x from almost all sign-and-support patterns of e. This suggests that accurate and efficient recovery of sparse signals is possible even with nearly 100% of the observations corrupted.