Support recovery of sparse signals from noisy measurements with orthogonal matching pursuit (OMP) has been extensively studied. In this paper, we show that for any $K$ -sparse signal ${\boldsymbol{x}}$ , if a sensing matrix $\boldsymbol{A}$ satisfies the restricted isometry property (RIP) with restricted isometry constant $\delta _{K+1} , then under some constraints on the minimum magnitude of nonzero elements of ${\boldsymbol{x}}$ , OMP exactly recovers the support of ${\boldsymbol{x}}$ from its measurements ${\boldsymbol{y}}=\boldsymbol{A}{\boldsymbol{x}}+\boldsymbol{v}$ in $K$ iterations, where $\boldsymbol{v}$ is a noise vector that is $\ell _2$ or $\ell _{\infty }$ bounded. This sufficient condition is sharp in terms of $\delta _{K+1}$ since for any given positive integer $K$ and any $1/\sqrt{K+1}\leq \delta , there always exists a matrix $\boldsymbol{A}$ satisfying the RIP with $\delta _{K+1}=\delta$ for which OMP fails to recover a $K$ -sparse signal ${\boldsymbol{x}}$ in $K$ iterations. Also, our constraints on the minimum magnitude of nonzero elements of ${\boldsymbol{x}}$ are weaker than existing ones. Moreover, we propose worst case necessary conditions for the exact support recovery of ${\boldsymbol{x}}$ , characterized by the minimum magnitude of the nonzero elements of ${\boldsymbol{x}}$ .

A Sharp Condition for Exact Support Recovery With Orthogonal Matching Pursuit

A sufficient condition reported very recently for perfect recovery of a K-sparse vector via orthogonal matching pursuit (OMP) in K iterations (when there is no noise) is that the restricted isometry constant (RIC) of the sensing matrix satisfies δ
 K+1
 <; (1/√(K) + 1) In the noisy case, this RIC upper bound along with a requirement on the minimal signal entry magnitude is known to guarantee exact support identification In this paper, we show that, in the presence of noise, a relaxed RIC upper bound δ
 K+1
 <; (√(4K + 1) - 1/2K) together with a relaxed requirement on the minimal signal entry magnitude suffices to achieve perfect support identification using OMP In the noiseless case, our result asserts that such a relaxed RIC upper bound can ensure exact support recovery in K iterations: this narrows the gap between the so far best known bound δ
 K+1
 <; (1/√(K( + 1)) and the ultimate performance guarantee δ
 K+1
 = (1/(K)) Our approach relies on a newly established near orthogonality condition, characterized via the achievable angles between two orthogonal sparse vectors upon compression, and, thus, better exploits the knowledge about the geometry of the compressed space The proposed near orthogonality condition can be also exploited to derive less restricted sufficient conditions for signal reconstruction in two other compressive sensing problems, namely, compressive domain interference cancellation and support identification via the subspace pursuit algorithm

https://ir.nctu.edu.tw:443/bitstream/11536/25189/1/000342415600040.pdf

An Improved RIP-Based Performance Guarantee for Sparse Signal Recovery via Orthogonal Matching Pursuit

Support recovery of sparse signals from compressed linear measurements is a fundamental problem in compressed sensing (CS). In this article, we study the orthogonal matching pursuit (OMP) algorithm for the recovery of support under noise. We consider two signal-to-noise ratio (SNR) settings: 1) the SNR depends on the sparsity level $K$ of input signals, and 2) the SNR is an absolute constant independent of $K$ . For the first setting, we establish necessary and sufficient conditions for the exact support recovery with OMP, expressed as lower bounds on the SNR. Our results indicate that in order to ensure the exact support recovery of all $K$ -sparse signals with the OMP algorithm, the SNR must at least scale linearly with the sparsity level $K$ . In the second setting, since the necessary condition on the SNR is not fulfilled, the exact support recovery with OMP is impossible. However, our analysis shows that recovery with an arbitrarily small but constant fraction of errors is possible with the OMP algorithm. This result may be useful for some practical applications where obtaining some large fraction of support positions is adequate.

Support Recovery With Orthogonal Matching Pursuit in the Presence of Noise

There is much discussion in industry and academia about possible technical solutions to address the growth in demand for wireless broadband. Massive multiple-input multiple-output (MIMO) systems are one of the most popular solutions to addressing this broadband demand in fifth generation (5G) cellular systems. Massive MIMO systems employ tens or hundreds of antennas at the base station to enable advanced multiuser MIMO communications. To reap the massive MIMO throughput gain, coherent transmission exploiting accurate channel state information at the transmitter is required. While it is expected that many 5G systems will employ frequency division duplexing (FDD), channel sounding for FDD systems requires a large pilot overhead, which usually scales proportionally to the number of transmit antennas. To resolve this problem, a compressed sensing (CS)-aided channel estimation scheme is proposed, which exploits the observation that the channel statistics change slowly in time. By utilizing a conventional least squares approach and a CS technique simultaneously, the proposed scheme reduces the pilot overhead. Simulation results show that the proposed scheme can estimate the channel with a reduced pilot overhead even when conventional CS cannot be applied.

Compressed Sensing-Aided Downlink Channel Training for FDD Massive MIMO Systems

In millimeter wave (mmWave) communication systems, large antenna arrays are used to compensate high path loss. While the large array provides high beamforming gain, it also poses a challenge in channel estimation. Since mmWave channels are likely to be sparse in angular domain, the channel estimation can be converted into a sparse recovery problem, and compressed sensing (CS) can be leveraged for the channel estimation. However, conventional non-adaptive CS algorithms show poor recovery performance with low signal-to-noise ratio (SNR), which is common before beamforming in mmWave channels. Although recently developed adaptive CS schemes perform better in a low SNR regime, their excessive feedback requirement hinders practical usage. In this paper, we propose a two-stage CS scheme that requires one-time feedback and is robust to noise, which can be regarded as a compromise between the two approaches. Sufficient conditions for the support recovery with the proposed scheme are characterized, and its effectiveness is also shown numerically.

Two-stage compressed sensing for millimeter wave channel estimation

Orthogonal matching pursuit (OMP) algorithm is a classical greedy algorithm in Compressed Sensing. In this letter, we study the performance of OMP in recovering the support of a sparse signal from a few noisy linear measurements. We consider two types of bounded noise and our analysis is in the framework of restricted isometry property (RIP). It is shown that under some conditions on RIP and the minimum magnitude of the nonzero elements of the sparse signal, OMP with proper stopping rules can recover the support of the signal exactly from the noisy observation. We also discuss the case of Gaussian noise. Our conditions on RIP improve some existing results.

Di-Rong Chen

Papers

The Exact Support Recovery of Sparse Signals With Noise via Orthogonal Matching Pursuit