Information-theoretic bounds on sparsity recovery in the high-dimensional and noisy setting

The problem of recovering the sparsity pattern of a fixed but unknown vector beta* and R^p based on a set of n noisy observations arises in a variety of settings, including subset selection in regression, graphical model selection, signal denoising, compressive sensing, and constructive approximation. Of interest are conditions on the model dimension p, the sparsity index s (number of non-zero entries in beta*), and the number of observations n that are necessary and/or sufficient to ensure asymptotically perfect recovery of the sparsity pattern. This paper focuses on the information-theoretic limits of sparsity recovery: in particular, for a noisy linear observation model based on measurement vectors drawn from the standard Gaussian ensemble, we derive both a set of sufficient conditions for asymptotically perfect recovery using the optimal decoder, as well as a set of necessary conditions that any decoder must satisfy for perfect recovery. This analysis of optimal decoding limits complements our previous work on thresholds for the behavior of l₁ -constrained quadratic programming for Gaussian measurement ensembles.