Sparsity and incoherence in compressive sampling

Abstract

We consider the problem of reconstructing a sparse signal x^0\in{\bb R}^n from a limited number of linear measurements. Given m randomly selected samples of Ux0, where U is an orthonormal matrix, we show that ell1 minimization recovers x0 exactly when the number of measurements exceeds m\geq \mathrm{const}^{\vphantom{\frac12}}_{\vphantom{\frac12}} \cdot\mu^2(U)\cdot S\cdot\log n, where S is the number of nonzero components in x0 and μ is the largest entry in U properly normalized: \mu(U) = \sqrt{n} \cdot \max_{k,j} |U_{k,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.