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In compressed sensing, the measurement is usually contaminated by additivenoise, and hence the information on the noise variance is often required todesign algorithms. In this paper, we propose an estimation method for theunknown noise variance in compressed sensing problems. The proposed method,called asymptotic residual matching (ARM), estimates the noise variance from asingle measurement vector on the basis of the asymptotic result for the$\ell_{1}$ optimization problem. Specifically, we derive the asymptoticresidual corresponding to the $\ell_{1}$ optimization and show that it dependson the noise variance. The proposed ARM approach obtains the estimate bycomparing the asymptotic residual with the actual one, which can be obtained bythe empirical reconstruction without the information of the noise variance. Forthe proposed ARM, we also propose a method to choose a reasonable parameter onthe basis of the asymptotic residual as well. The idea of the proposed ARM canbe applied not only to the reconstruction of the sparse vector but also to thatof other structured vectors such as the binary vector. Simulation results showthat the proposed noise variance estimation outperforms several conventionalmethods, especially when the problem size is small. We also show that, by usingthe proposed method, we can tune the regularization parameter of the $\ell_{1}$optimization to achieve good reconstruction performance even when the noisevariance is unknown.

Noise Variance Estimation Using Asymptotic Residual in Compressed Sensing