APSIPA Transactions on Signal and Information Processing > Vol 12 > Issue 1

Noise Variance Estimation Using Asymptotic Residual in Compressed Sensing

Ryo Hayakawa, Institute of Engineering, Tokyo University of Agriculture and Technology, Japan, hayakawa@go.tuat.ac.jp
Suggested Citation
Ryo Hayakawa (2023), "Noise Variance Estimation Using Asymptotic Residual in Compressed Sensing", APSIPA Transactions on Signal and Information Processing: Vol. 12: No. 1, e46. http://dx.doi.org/10.1561/116.00000215

Publication Date: 13 Nov 2023
© 2023 R. Hayakawa
Compressed sensingnoise variance estimationconvex optimizationasymptotic analysis


Open Access

This is published under the terms of CC BY-NC.

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In this article:
Noise Variance Estimation in Compressed Sensing 
Related Work 
Asymptotic Residual for ℓ1 Optimization 
Proposed Noise Variance Estimation 
Extension to Other Structured Vectors 
Simulation Results 
Conclusion and Future Work 


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 ℓ1 optimization problem. Specifically, we derive the asymptotic residual corresponding to the ℓ1 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 ℓ1 optimization to achieve good reconstruction performance, even when the noise variance is unknown.