Asymptotic Analysis of MAP Estimation via the Replica Method and Applications to Compressed Sensing
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Goyal_Asymptotic analysis.pdf
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Author(s)
Goyal, Vivek K.
Rangan, Sundeep
Fletcher, Alyson K.
Date Issued
March 2012
Journal
IEEE Transactions on Information Theory
Publisher
Institute of Electrical and Electronics Engineers (IEEE)
Citation
Rangan, Sundeep, Alyson K. Fletcher, and Vivek K Goyal. “Asymptotic Analysis of MAP Estimation via the Replica Method and Applications to Compressed Sensing.” IEEE Transactions on Information Theory 58.3 (2012): 1902–1923.
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Author's final manuscript
Abstract
The replica method is a nonrigorous but well-known technique from statistical physics used in the asymptotic analysis of large, random, nonlinear problems. This paper applies the replica method, under the assumption of replica symmetry, to study estimators that are maximum a posteriori (MAP) under a postulated prior distribution. It is shown that with random linear measurements and Gaussian noise, the replica-symmetric prediction of the asymptotic behavior of the postulated MAP estimate of an -dimensional vector “decouples” as scalar postulated MAP estimators. The result is based on applying a hardening argument to the replica analysis of postulated posterior mean estimators of Tanaka and of Guo and Verdú. The replica-symmetric postulated MAP analysis can be readily applied to many estimators used in compressed sensing, including basis pursuit, least absolute shrinkage and selection operator (LASSO), linear estimation with thresholding, and zero norm-regularized estimation. In the case of LASSO estimation, the scalar estimator reduces to a soft-thresholding operator, and for zero norm-regularized estimation, it reduces to a hard threshold. Among other benefits, the replica method provides a computationally tractable method for precisely predicting various performance metrics including mean-squared error and sparsity pattern recovery probability.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Research Laboratory of Electronics
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DOI of Published Version
https://doi.org/10.1109/tit.2011.2177575
