Robust Nonnegative Sparse Recovery and the Nullspace Property of 0/1 Measurements

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 <inline-formula> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula>-regularization is required. As an important proof of principle, we establish that <inline-formula> <tex-math notation="LaTeX">$m\times n$ </tex-math></inline-formula> random i.i.d. 0/1-valued Bernoulli matrices obey the required conditions with overwhelming probability provided that <inline-formula> <tex-math notation="LaTeX">$m= \mathcal {O}(s\log (n/s))$ </tex-math></inline-formula>. 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.