Sharp thresholds for high-dimensional and noisy recovery of sparsity

TL;DR

This paper establishes precise thresholds for the number of observations needed for the Lasso method to reliably recover the sparsity pattern of a high-dimensional, noisy signal, highlighting a phase transition in recovery success.

Contribution

The paper derives explicit sharp thresholds for sparse recovery using Lasso in high-dimensional Gaussian models, clarifying the exact conditions for success and failure.

Findings

Recovery thresholds depend on problem dimension and sparsity level.

For Gaussian ensembles, the thresholds are exactly determined as 1.

Lasso succeeds or fails with high probability depending on the number of observations relative to thresholds.

Abstract

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 $\ell_1$-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 $\epsilon > 0$, 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.