The problem of consistently estimating the sparsity pattern of a vector
\betastar \in \real^\mdim 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 \mdim, the
number \spindex of non-zero elements in \betastar, and the number of
observations \numobs 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 \ThreshLow and
\ThreshUp with the following properties: for any ϵ>0, if \numobs
> 2 (\ThreshUp + \epsilon) \log (\mdim - \spindex) + \spindex + 1, then the
Lasso succeeds in recovering the sparsity pattern with probability converging
to one for large problems, whereas for \numobs < 2 (\ThreshLow - \epsilon)
\log (\mdim - \spindex) + \spindex + 1, then the probability of successful
recovery converges to zero. For the special case of the uniform Gaussian
ensemble, we show that \ThreshLow = \ThreshUp = 1, so that the threshold is
sharp and exactly determined.Comment: Appeared as Technical Report 708, Department of Statistics, UC
Berkele