Sparsity and incoherence in compressive sampling

We consider the problem of reconstructing a sparse signal from a limited number of linear measurements. Given m randomly selected samples of Ux 0 , where U is an orthonormal matrix, we show that ℓ 1 minimization recovers x 0 exactly when the number of measurements exceeds where S is the number of nonzero components in x 0 and μ is the largest entry in U properly normalized: . The smaller μ is, the fewer samples needed. The result holds for ‘most’ sparse signals x 0 supported on a fixed (but arbitrary) set T . Given T , if the sign of x 0 for each nonzero entry on T and the observed values of Ux 0 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.