Performance tradeoffs for exact support recovery of sparse signals

Abstract

Abstract-We study the tradeoffs between the number of measurements, the signal sparsity level, and the measurement noise level for exact support recovery of sparse signals via random noisy measurements. By drawing analogy between exact support recovery and communication over the Gaussian multiple access channel, and exploiting mathematical tools developed for the latter problem, we derive sharp asymptotic sufficient and necessary conditions for exact support recovery. Specifically, when the number of nonzero entries is held fixed, the exact asymptotics on the number of measurements for support recovery is developed. When the number of nonzero entries increases in certain manners, we obtain sufficient conditions tighter than existing results. The proposed information theoretic framework for analyzing the performance of support recovery is further demonstrated to be capable of dealing with a variety of sparse signal recovery models.

An achievable measurement rate-MSE tradeoff in compressive sensing through partial support recovery

“…We have established sufficient conditions on the measurement rate to ensure partial detection of the support set based on the relative power of the non-zero entries in the sparse signal. This builds on and extends the results in [7,8,9], which considered only perfect recovery of the support set.…”

“…The asymptotic tradeoffs for exact support set recovery in a noisy setting were studied in [7,8,9]. Further, asymptotically achievable Cramér-Rao bounds on the mean-square estimation error (MSE) were given in [10,11,12].…”

“…In this paper, we adopt the approach of [7,8,9] and derive achievable performance guarantees for partial support set recovery. We use the result to derive an achievable tradeoff between the mean square error and the measurement rate, defined as a relation between the dimensions of the measurement matrix.…”

On the benefits of the block-sparsity structure in sparse signal recovery

“…In the noiseless environment (i.e., Z = 0), sufficient conditions to exactly recover the support of the sparse signal have been derived in [3]- [5]. In the presence of measurement noise, information theoretic tools have proven useful in understanding the performance tradeoff for support recovery of sparse signals [6]- [10]. In particular, Jin et al [10] identified the connection between the problem of sparse signal support recovery and the problem of communication over Gaussian multiple access channel (MAC), based on which they derived sharper asymptotic tradeoffs between the signal dimension, the number of nonzero entries, and the number of measurements for exact support recovery in the noisy setting.…”

“…In addition, so called, diversity effect is discussed through which the block-sparsity can further reduce the minimum number of measurements, especially in the regime of moderate and low SNR. Our work can be viewed as a generalization of the work in [10] in the sense that we interpret our problem as the problem of communications over Gaussian multi-input and single-output (MISO) MAC.…”