Stability and Instance Optimality for Gaussian Measurements in Compressed Sensing
Foundations of Computational Mathematics
Decoding by linear programming
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
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Compressed sensing aims at recovering a sparse signal $x\in \mathbb{R}^N$ from few nonadaptive, linear measurements $\Phi(x)$ given by a measurement matrix $\Phi$. One of the fundamental recovery algorithms is an $\ell_1$ minimization. In this paper we investigate the situation when our measurement $\Phi(x)$ is contaminated by arbitrary noise under the assumption that the matrix $\Phi$ satisfies the restricted isometry property. This complements results from [Candès, Romberg, and Tao, Comm. Pure Appl. Math., 59 (2006), pp. 1207-1223] and [DeVore, Petrova, and Wojtaszczyk, Appl. Comput. Harmon. Anal., 27 (2009), pp. 275-288].