The fundamental limits of stable recovery in compressed sensing

Compressed sensing has shown that a wide variety of structured signals can be recovered from a limited number of noisy linear measurements. This paper considers the extent to which such recovery is robust to signal and measurement uncertainty. The main result is a non-asymptotic upper bound on the reconstruction error in terms of two key quantities: the best approximation error of the signal (with respect to a user-defined approximation set) and the measurement error. We assume a random Gaussian sensing matrix but place no restrictions on the signal or the noise. This result provides a simple and yet powerful framework for analyzing the fundamental limits of stable recovery, allowing us to sharpen existing results as well as derive new ones.