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Asymptotic analysis of MAP estimation via the replica method and applications to compressed sensing (2009)
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BibTeX
@MISC{Rangan09asymptoticanalysis,
author = {Sundeep Rangan and Alyson K. Fletcher and Vivek K Goyal},
title = {Asymptotic analysis of MAP estimation via the replica method and applications to compressed sensing },
year = {2009}
}
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Abstract
The replica method is a non-rigorous but widely-accepted technique from statistical physics used in the asymptotic analysis of large, random, nonlinear problems. This paper applies the replica method to non-Gaussian maximum a posteriori (MAP) estimation. It is shown that with random linear measurements and Gaussian noise, the asymptotic behavior of the MAP estimate of ann-dimensional vector “decouples ” asnscalar MAP estimators. The result is a counterpart to Guo and Verdú’s replica analysis of minimum mean-squared error estimation. The replica MAP analysis can be readily applied to many estimators used in compressed sensing, including basis pursuit, 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 normregularized estimation it reduces to a hard-threshold. Among other benefits, the replica method provides a computationally-tractable method for exactly computing various performance metrics including mean-squared error and sparsity pattern recovery probability.
Keyphrases
replica method asymptotic analysis map estimation replica analysis minimum mean-squared error estimation nonlinear problem lasso estimation norm-regularized estimation many estimator gaussian noise compressed sensing various performance metric mean-squared error map estimate normregularized estimation statistical physic linear estimation widely-accepted technique random linear measurement scalar estimator reduces asymptotic behavior basis pursuit sparsity pattern recovery probability replica map analysis soft-thresholding operator non-gaussian maximum computationally-tractable method
