Stable Recovery and Separation of Structured Signals by Convex Geometry

We consider the ill-posed problem of identifying the signal that is structured in some general dictionary (i.e., possibly redundant or over-complete matrix) from corrupted measurements, where the corruption is structured in another general dictionary. We formulate the problem by applying appropriate convex constraints to the signal and corruption according to their structures and provide conditions for exact recovery from structured corruption and stable recovery from structured corruption with added stochastic bounded noise. In addition, this paper provides estimates of the number of the measurements needed for recovery. These estimates are based on computing the Gaussian complexity of a tangent cone and the Gaussian distance to a subdifferential. Applications covered by the proposed programs include the recovery of signals that is disturbed by impulse noise, missing entries, sparse outliers, random bounded noise, and signal separation. Numerical simulation results are presented to verify and complement the theoretical results.