Noise Variance Estimation Using Asymptotic Residual in Compressed Sensing

TL;DR

This paper introduces a novel method called asymptotic residual matching (ARM) for accurately estimating noise variance in compressed sensing, enabling improved algorithm tuning and reconstruction performance from a single measurement vector.

Contribution

The paper proposes the ARM method that leverages asymptotic analysis of the residual in  optimization to estimate noise variance without multiple measurements or prior noise information.

Findings

ARM outperforms conventional noise estimation methods in small problem sizes.

Using ARM improves regularization parameter tuning for  optimization.

The method enhances reconstruction accuracy when noise variance is unknown.

Abstract

In compressed sensing, measurements are typically contaminated by additive noise, and therefore, information about the noise variance is often needed to design algorithms. In this paper, we propose a method for estimating 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 empirical reconstruction without the information on the noise variance. For the proposed ARM, we also propose a method to choose a reasonable parameter based on the asymptotic residual. Simulation results show that the proposed noise variance estimation outperforms several conventional methods, especially when the problem size is small. We also show that, by using the proposed method, we can tune the regularization parameter of the $\ell_{1}$ optimization to achieve good reconstruction performance, even when the noise variance is unknown.