The Failure Case of Phase Transition for Penalized Problems in Corrupted Sensing

Abstract

The problem of corrupted sensing aims to recover a signal from corrupted measurements. As a result of the ill-posedness of the problem, recovery is impossible for arbitrary signal and corruption. On the contrary, if both the signal and corruption are well structured, recovery becomes possible, and a common approach is to solve a penalized program. In practice, penalized programs exhibit a sharp phase transition, and this letter presents a partial analysis for this phenomenon, which focuses on the problem when they fail. In the analysis, we present a geometry condition determining the failure of penalized programs. Then, if the sensing matrix has independent normal entries, the geometry condition can be studied using Gaussian process theory, and we obtain the threshold for the measurement number below which penalized problems fail with high probability. Moreover, to compute the obtained threshold in practice, we present a computable upper bound for it. Simulation results demonstrate that our theoretical results are very sharp.