Noise Variance Estimation Using Asymptotic Residual in Compressed Sensing

In compressed sensing, the measurement is usually contaminated by additive noise, and hence the information of the noise variance is often required to design algorithms. In this paper, we propose an estimation method for the unknown noise variance in compressed sensing problems. The proposed method called asymptotic residual matching (ARM) estimates the noise variance from a single measurement vector on the basis of the asymptotic result for the $\ell_{1}$ optimization problem. Specifically, we derive the asymptotic residual corresponding to the $\ell_{1}$ optimization and show that it depends on the noise variance. The proposed ARM approach obtains the estimate by comparing the asymptotic residual with the actual one, which can be obtained by the empirical reconstruction without the information of the noise variance. Simulation results show that the proposed noise variance estimation outperforms a conventional method based on the analysis of the ridge regularized least squares. We also show that, by using the proposed method, we can achieve good reconstruction performance in compressed sensing even when the noise variance is unknown.