Dense Error Correction via L1-Minimization

This paper studies the problem of recovering a non-negative sparse signal $\x \in \Re^n$ from highly corrupted linear measurements $\y = A\x + \e \in \Re^m$, where $\e$ is an unknown error vector whose nonzero entries may be unbounded. Motivated by an observation from face recognition in computer vision, this paper proves that for highly correlated (and possibly overcomplete) dictionaries $A$, any non-negative, sufficiently sparse signal $\x$ can be recovered by solving an $\ell^{1$-minimization} problem: $\min |\x|1 + |\e|1 \quad {subject to} \quad \y = A\x + \e.$ More precisely, 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, then as $m$ goes to infinity, the above $\ell^{1$-minimization} succeeds for all signals $\x$ and almost all sign-and-support patterns of $\e$. This result suggests that accurate recovery of sparse signals is possible and computationally feasible even with nearly 100% of the observations corrupted. The proof relies on a careful characterization of the faces of a convex polytope spanned together by the standard crosspolytope and a set of iid Gaussian vectors with nonzero mean and small variance, which we call the ``cross-and-bouquet'' model. Simulations and experimental results corroborate the findings, and suggest extensions to the result.