The noise collector for sparse recovery in high dimensions

The ability to detect sparse signals from noisy, high-dimensional data is a top priority in modern science and engineering. It is well known that a sparse solution of the linear system A_ρ = b₀ can be found efficiently with an l₁-norm minimization approach if the data are noiseless. However, detection of the signal from data corrupted by noise is still a challenging problem as the solution depends, in general, on a regularization parameter with optimal value that is not easy to choose. We propose an efficient approach that does not require any parameter estimation. We introduce a no-phantom τ and the Noise Collector matrix C and solve an augmented system A_ρ + Cn = b₀ + e, where e is the noise. We show that the l₁-norm minimal solution of this system has zero false discovery rate for any level of noise, with probability that tends to one as the dimension of b₀ increases to infinity. We obtain exact support recovery if the noise is not too large and develop a fast Noise Collector algorithm, which makes the computational cost of solving the augmented system comparable with that of the original one. We demonstrate the effectiveness of the method in applications to passive array imaging.