High-dimensional subset recovery in noise: Sparsified measurements without loss of statistical efficiency

We consider the problem of estimating the support of a vector β ∈ R based on observations contaminated by noise. A significant body of work has studied behavior of l1-relaxations when applied to measurement matrices drawn from standard dense ensembles (e.g., Gaussian, Bernoulli). In this paper, we analyze sparsified measurement ensembles, and consider the tradeoff between measurement sparsity, as measured by the fraction γ of non-zero entries, and the statistical efficiency, as measured by the minimal number of observations n required for exact support recovery with probability converging to one. Our main result is to prove that it is possible to let γ → 0 at some rate, yielding measurement matrices with a vanishing fraction of non-zeros per row while retaining the same statistical efficiency as dense ensembles. A variety of simulation results confirm the sharpness of our theoretical predictions.