Uniform Recovery Bounds for Structured Random Matrices in Corrupted Compressed Sensing - 2018


We study the problem of recovering an s-sparse signal x* ? C n from corrupted measurements y = Ax* + z* + w, where z* ? C m is a k-sparse corruption vector whose nonzero entries could be arbitrarily giant and w ? C mis a dense noise with bounded energy. The aim is to exactly and stably recover the sparse signal with tractable optimization programs. During this Project, we have a tendency to prove the uniform recovery guarantee of this drawback for two classes of structured sensing matrices. The primary class can be expressed as the merchandise of a unit-norm tight frame (UTF), a random diagonal matrix, and a bounded columnwise orthonormal matrix (e.g., partial random circulant matrix). When the UTF is bounded (i.e. µ(U) ~ 1/vm), we prove that with high probability, one can recover an s-sparse signal precisely and stably by l one minimization programs whether or not the measurements are corrupted by a sparse vector, provided m = O(s log 2 s log two n) and therefore the sparsity level k of the corruption could be a constant fraction of the total variety of measurements. The second category considers a randomly subsampled orthonormal matrix (e.g., random Fourier matrix). We have a tendency to prove the uniform recovery guarantee provided that the corruption is sparse on bound sparsifying domain. Numerous simulation results also are presented to verify and complement the theoretical results.

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