On sparse reconstruction from Fourier and Gaussian measurements

This paper improves upon best‐known guarantees for exact reconstruction of a sparse signal f from a small universal sample of Fourier measurements. The method for reconstruction that has recently gained momentum in the sparse approximation theory is to relax this highly nonconvex problem to a convex problem and then solve it as a linear program. We show that there exists a set of frequencies Ω such that one can exactly reconstruct every r ‐sparse signal f of length n from its frequencies in Ω, using the convex relaxation, and Ω has size A random set Ω satisfies this with high probability. This estimate is optimal within the log log n and log 3 r factors. We also give a relatively short argument for a similar problem with k ( r, n ) ≈ r [12 + 8 log( n / r )] Gaussian measurements. We use methods of geometric functional analysis and probability theory in Banach spaces, which makes our arguments quite short.