  We investigate recovery of nonnegative vectors from non-adaptive compressive
measurements in the presence of noise of unknown power. In the absence of
noise, existing results in the literature identify properties of the
measurement that assure uniqueness in the non-negative orthant. By linking such
uniqueness results to nullspace properties, we deduce uniform and robust
compressed sensing guarantees for nonnegative least squares. No
l1-regularization is required. As an important proof of principle, we establish
that m x n random i.i.d. 0/1-valued Bernoulli matrices obey the required
conditions with overwhelming probability provided that m=O(slog(n/s)). We
achieve this by establishing the robust nullspace property for random
0/1-matrices - a novel result in its own right. Our analysis is motivated by
applications in wireless network activity detection.
