Sparse Non-Negative Recovery from Biased Subgaussian Measurements using NNLS

We investigate non-negative least squares (NNLS) for the recovery of sparse non-negative vectors from noisy linear and biased measurements. We build upon recent results from [1] showing that for matrices whose row-span intersects the positive orthant, the nullspace property (NSP) implies compressed sensing recovery guarantees for NNLS. Such results are as good as for l_1-regularized estimators but require no tuning at all. A bias in the sensing matrix improves this auto-regularization feature of NNLS and the NSP then determines the sparse recovery performance only. We show that NSP holds with high probability for shifted symmetric subgaussian matrices and its quality is independent of the bias. As tool for proving this result we established a debiased version of Mendelson's small ball method.