Necessary and Sufficient Conditions for Sparsity Pattern Recovery

Abstract

The problem of detecting the sparsity pattern of a k-sparse vector in R n from m random noisy measurements is of interest in many areas such as system identification, denoising, pattern recognition, and compressed sensing. This paper addresses the scaling of the number of measurements m, with signal dimension n and sparsity-level nonzeros k, for asymptotically-reliable detection. We show a necessary condition for perfect recovery at any given SNR for all algorithms, regardless of complexity, is m = Ω(k log(n − k)) measurements. Conversely, it is shown that this scaling of Ω(k log(n − k)) measurements is sufficient for a remarkably simple "maximum correlation" estimator. Hence this scaling is optimal and does not require more sophisticated techniques such as lasso or matching pursuit. The constants for both the necessary and sufficient conditions are precisely defined in terms of the minimum-toaverage ratio of the nonzero components and the SNR. The necessary condition improves upon previous results for maximum likelihood estimation. For lasso, it also provides a necessary condition at any SNR and for low SNR improves upon previous work. The sufficient condition provides the first asymptotically-reliable detection guarantee at finite SNR.

Performance tradeoffs for exact support recovery of sparse signals

“…Sufficient and necessary conditions are developed for different sparsity levels. Using the same decoder, Fletcher et al [19] further improved the necessary condition in certain settings. Wang et al [20] also presented a set of necessary conditions for exact support recovery.…”

Support Recovery With Sparsely Sampled Free Random Matrices

“…Recovery of the sparsity pattern with vanishing error probability is studied in a number of recent works such as [1], [2], [14], [27], [39], [40]. When k = n i=1 b i , the number of nonzero coefficients in v, is known beforehand 2 and their magnitude is bounded away from zero, exact support recovery requires that the number of measurements grow as k log n [14], [40].…”

“…When k = n i=1 b i , the number of nonzero coefficients in v, is known beforehand 2 and their magnitude is bounded away from zero, exact support recovery requires that the number of measurements grow as k log n [14], [40]. If the support recovery error rate is allowed to be non-vanishing, fewer measurements are necessary.…”

Reduced-dimension multiuser detection

“…The model (11) for RD-MUD has a similar form to the observation model in the compressed sensing literature [13], [22], except that the noise in the RD-MUD front-end output is colored. Hence, to recover b, we can combine ideas developed in the context of compressed sensing and MUD.…”

“…While such sparsity has been exploited in various detection settings, there is still a gap in applying these ideas to the multiuser setting we consider here. Most existing work on exploiting compressed sensing [11], [12] for signal detection assumes discrete signals and then applies compressed sensing via matrix multiplication [8], [13], [14], [15], [16]. In contrast, in multiuser detection the received signal is continuous.…”