Asymptotic analysis of MAP estimation via the replica method and applications to compressed sensing

Sundeep Rangan, Alyson K. Fletcher, Vivek K. Goyal

Research output: Contribution to journalArticlepeer-review


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 n-dimensional vector "decouples" as n 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 Verd. 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.

Original languageEnglish (US)
Article number6157073
Pages (from-to)1902-1923
Number of pages22
JournalIEEE Transactions on Information Theory
Issue number3
StatePublished - Mar 2012


  • Compressed sensing
  • Laplace's method
  • large deviations
  • least absolute shrinkage and selection operator (LASSO)
  • non-Gaussian estimation
  • nonlinear estimation
  • random matrices
  • sparsity
  • spin glasses
  • statistical mechanics
  • thresholding

ASJC Scopus subject areas

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


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