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We study the problem of recovering an <inline-formula><tex-math notation="LaTeX">$s$</tex-math></inline-formula> -sparse signal <inline-formula><tex-math notation="LaTeX">$\mathbf{x}^{\star }\in \mathbb {C}^n$</tex-math> </inline-formula> from corrupted measurements <inline-formula><tex-math notation="LaTeX"> $\mathbf{y}=\mathbf{A}\mathbf{x}^{\star }+\mathbf{z}^{\star }+\mathbf{w}$</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">$\mathbf{z}^{\star }\in \mathbb {C}^m$</tex-math></inline-formula> is a <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-sparse corruption vector whose nonzero entries may be arbitrarily large and <inline-formula><tex-math notation="LaTeX">$\mathbf{w}\in \mathbb {C}^m$ </tex-math></inline-formula> is a dense noise with bounded energy. The aim is to exactly and stably recover the sparse signal with tractable optimization programs. In this paper, we prove the <italic>uniform recovery guarantee</italic> of this problem for two classes of structured sensing matrices. The first class can be expressed as the product 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. <inline-formula><tex-math notation="LaTeX">$\mu (\mathbf{U})\sim 1/\sqrt{m}$</tex-math></inline-formula>), we prove that with high probability, one can recover an <inline-formula><tex-math notation="LaTeX">$s$</tex-math></inline-formula>-sparse signal exactly and stably by <inline-formula><tex-math notation="LaTeX">$l_1$</tex-math></inline-formula> minimization programs even if the measurements are corrupted by a sparse vector, provided <inline-formula><tex-math notation="LaTeX"> $m=\mathcal{O}(s\log ^2s\log ^2n)$</tex-math></inline-formula> and the sparsity level <inline-formula> <tex-math notation="LaTeX">$k$</tex-math></inline-formula> of the corruption is a constant fraction of the total number of measurements. The second class considers a randomly subsampled orthonormal matrix (e.g., random Fourier matrix). We prove the uniform recovery guarantee provided that the corruption is sparse on certain sparsifying domain. Numerous simulation results are also presented to verify and complement the theoretical results.

Uniform Recovery Bounds for Structured Random Matrices in Corrupted Compressed Sensing