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^* ∈ Rᵖ based on observations contaminated by noise. A significant body of work has studied behavior of ₁-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 trade-off 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.