Sparsity and incoherence in compressive sampling

We consider the problem of reconstructing a sparse signal x0 ∈ R from a limited number of linear measurements. Given m randomly selected samples of Ux0, where U is an orthonormal matrix, we show that 1 minimization recovers x0 exactly when the number of measurements exceeds m const · μ(U) · S · log n, where S is the number of nonzero components in x0 and μ is the largest entry in U properly normalized: μ(U) = √n · maxk,j |Uk,j |. The smaller μ is, the fewer samples needed. The result holds for ‘most’ sparse signals x0 supported on a fixed (but arbitrary) set T. Given T, if the sign of x0 for each nonzero entry on T and the observed values of Ux0 are drawn at random, the signal is recovered with overwhelming probability. Moreover, there is a sense in which this is nearly optimal since any method succeeding with the same probability would require just about as many samples.