  The ability to detect sparse signals from noisy high-dimensional data is a
top priority in modern science and engineering. A sparse solution of the linear
system $A \rho = b_0$ can be found efficiently with an $l_1$-norm minimization
approach if the data is noiseless. Detection of the signal's support from data
corrupted by noise is still a challenging problem, especially if the level of
noise must be estimated. We propose a new efficient approach that does not
require any parameter estimation. We introduce the Noise Collector (NC) matrix
$C$ and solve an augmented system $A \rho + C \eta = b_0 + e$, where $ e$ is
the noise. We show that the $l_1$-norm minimal solution of the augmented system
has zero false discovery rate for any level of noise and with probability that
tends to one as the dimension of $ b_0$ increases to infinity. We also 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 to that of the original one. Finally, we demonstrate the
effectiveness of the method in applications to passive array imaging.
