Sharp thresholds for high-dimensional and noisy recovery of sparsity using l1-constrained quadratic programming

The problem of consistently estimating the sparsity pattern of a vector β∗ ∈ R based on observations contaminated by noise arises in various contexts, including signal denoising, sparse approximation, compressed sensing, and model selection. We analyze the behavior of l1-constrained quadratic programming (QP), also referred to as the Lasso, for recovering the sparsity pattern. Our main result is to establish precise conditions on the problem dimension p, the number k of non-zero elements in β∗, and the number of observations n that are necessary and sufficient for subset selection using the Lasso. For a broad class of Gaussian ensembles satisfying mutual incoherence conditions, we establish existence and compute explicit values of thresholds 0 < θl ≤ 1 ≤ θu < +∞ with the following properties: for any δ > 0, if n > 2 (θu + δ) k log(p− k), then the Lasso succeeds in recovering the sparsity pattern with probability converging to one for large problems, whereas for n < 2 (θl − δ) k log(p − k), then the probability of successful recovery converges to zero. For the special case of the uniform Gaussian ensemble, we show that θl = θu = 1, so that the precise threshold n = 2 k log(p− k) is exactly determined.