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

Abstract: The problem of recovering the sparsity pattern of a fixed but unknown vector $$\beta^* \in \real^p based on a set of$$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^*),</sup>andthenumberofobservations),</sup> and the number of observations nthatarenecessaryand/orsufficienttoensureasymptoticallyperfectrecoveryofthesparsitypattern.Thispaperfocusesontheinformation−theoreticlimitsofsparsityrecovery:inparticular,foranoisylinearobservationmodelbasedonmeasurementvectorsdrawnfromthestandardGaussianensemble,wederivebothasetofsufficientconditionsforasymptoticallyperfectrecoveryusingtheoptimaldecoder,aswellasasetofnecessaryconditionsthatanydecoder,regardlessofitscomputationalcomplexity,mustsatisfyforperfectrecovery.Thisanalysisofoptimaldecodinglimitscomplementsourpreviouswork(ARXIV:math.ST/0605740)onsharpthresholdsforsparsityrecoveryusingtheLasso( 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, regardless of its computational complexity, must satisfy for perfect recovery. This analysis of optimal decoding limits complements our previous work (ARXIV: math.ST/0605740) on sharp thresholds for sparsity recovery using the Lasso (\ell_1$-constrained quadratic programming) with Gaussian measurement ensembles.