Stability and Instance Optimality for Gaussian Measurements in Compressed Sensing

In compressed sensing we seek to gain information about vector x ∈ R from d << N nonadaptive linear measurements. Candes, Donoho, Tao et. al. ( see e.g. [2, 4, 8]) proposed to seek good approximation to x via `1 minimisation. In this paper we show that in the case of Gaussian measurements it recovers the signal well from inacurate measurements, thus improving result from [4]. We also show that with big probability it gives information comparable with best k term approximation in euclidean norm, k ∼ d/ ln N . This provides the first numerically friendly algorithm to do so, see [7].