Sharp thresholds for high-dimensional and noisy recovery of sparsity

The problem of consistently estimating the sparsity pattern of a vector β∗ ∈ R based on observations contaminated by noise arises in various contexts, including subset selection in regression, structure estimation in graphical models, sparse approximation, and signal denoising. 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 a sharp relation between the problem dimension p, the number s of non-zero elements in β∗, and the number of observations n that are required for reliable recovery. For a broad class of Gaussian ensembles satisfying mutual incoherence conditions, we establish existence and compute explicit values of thresholds θl and θu with the following properties: for any ν > 0, if n > 2 (θu+ν) log(p−s)+s+1, then the Lasso succeeds in recovering the sparsity pattern with probability converging to one for large problems, whereas for n < 2 (θl−ν) log(p−s)+s+1, 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 threshold is sharp and exactly determined.