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

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

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 an n-dimensional vector "decouples" as n scalar 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 norm-regularized 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.

Original languageEnglish (US)
Title of host publicationAdvances in Neural Information Processing Systems 22 - Proceedings of the 2009 Conference
Pages1545-1553
Number of pages9
StatePublished - 2009
Event23rd Annual Conference on Neural Information Processing Systems, NIPS 2009 - Vancouver, BC, Canada
Duration: Dec 7 2009Dec 10 2009

Publication series

NameAdvances in Neural Information Processing Systems 22 - Proceedings of the 2009 Conference

Other

Other23rd Annual Conference on Neural Information Processing Systems, NIPS 2009
CountryCanada
CityVancouver, BC
Period12/7/0912/10/09

ASJC Scopus subject areas

  • Information Systems

Fingerprint Dive into the research topics of 'Asymptotic analysis of MAP estimation via the replica method and compressed sensing'. Together they form a unique fingerprint.

  • Cite this

    Rangan, S., Fletcher, A. K., & Goyal, V. K. (2009). Asymptotic analysis of MAP estimation via the replica method and compressed sensing. In Advances in Neural Information Processing Systems 22 - Proceedings of the 2009 Conference (pp. 1545-1553). (Advances in Neural Information Processing Systems 22 - Proceedings of the 2009 Conference).