Sparsity and incoherence in compressive sampling

We consider the problem of reconstructing a sparse signal x(0) is an element of R(n) 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 m >= const . mu(2)(U) . S . log n, where S is the number of nonzero components in x(0) and mu is the largest entry in U properly normalized: mu(U) = root n . max(k), (j) |U(k, j)|. The smaller mu 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.