High-Dimensional Variable Selection with Sparse Random Projections: Measurement Sparsity and Statistical Efficiency

Abstract

We consider the problem of high-dimensional variable selection: given n noisy observations of a k-sparse vector β* ∈ Rp, estimate the subset of non-zero entries of β*. A significant body of work has studied behavior of l1-relaxations when applied to random measurement matrices that are dense (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 correct variable selection with probability converging to one. Our main result is to prove that it is possible to let the fraction on non-zero entries 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.