Sharp Thresholds for High-Dimensional and Noisy Sparsity Recovery Using $\ell _{1}$-Constrained Quadratic Programming (Lasso)

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Abstract
The problem of consistently estimating the sparsity pattern of a vector beta* isin Rp 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 nonzero elements in beta*, and the number of observations n that are necessary and sufficient for sparsity pattern recovery using the Lasso. We first analyze the case of observations made using deterministic design matrices and sub-Gaussian additive noise, and provide sufficient conditions for support recovery and linfin-error bounds, as well as results showing the necessity of incoherence and bounds on the minimum value. We then turn to the case of random designs, in which each row of the design is drawn from a N (0, Sigma) ensemble. For a broad class of Gaussian ensembles satisfying mutual incoherence conditions, we compute explicit values of thresholds 0 < thetasl(Sigma) les thetasu(Sigma) < +infin with the following properties: for any delta > 0, if n > 2 (thetasu + delta) klog (p- k), then the Lasso succeeds in recovering the sparsity pattern with probability converging to one for large problems, whereas for n < 2 (thetasl - delta)klog (p - k), then the probability of successful recovery converges to zero. For the special case of the uniform Gaussian ensemble (Sigma = Iptimesp), we show that thetasl = thetas<u = 1, so that the precise threshold n = 2 klog(p- k) is exactly determined.

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