Asymptotic MMSE analysis under sparse representation modeling

Wasim Huleihel, Neri Merhav

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Scopus citations


Compressed sensing is a signal processing technique in which data is acquired directly in a compressed form. There are two modeling approaches that can be considered: the worst-case (Hamming) approach and a statistical mechanism, in which the signals are modeled as random processes rather than as individual sequences. In this paper, the second approach is studied. Accordingly, we consider a model of the form Y = HX +W, where each component of X is given by Xi = SiUi, where {Ui} are i.i.d. Gaussian random variables, and {Si} are binary random variables independent of {Ui{, and not necessarily independent and identically distributed (i.i.d.), H ε ℝk×n is a random matrix with i.i.d. entries, and W is white Gaussian noise. Using a direct relationship between optimum estimation and certain partition functions, and by invoking methods from statistical mechanics and from random matrix theory, we derive an asymptotic formula for the minimum mean-square error (MMSE) of estimating the input vector X given Y and H, as k, n → ∞, keeping the measurement rate, R = k/n, fixed. In contrast to previous derivations, which are based on the replica method, the analysis carried in this paper is rigorous. In contrast to previous works in which only memoryless sources were considered, we consider a more general model which allows a certain structured dependency among the various components of the source.

Original languageEnglish
Title of host publication2014 IEEE International Symposium on Information Theory, ISIT 2014
PublisherInstitute of Electrical and Electronics Engineers Inc.
Number of pages5
ISBN (Print)9781479951864
StatePublished - 2014
Externally publishedYes
Event2014 IEEE International Symposium on Information Theory, ISIT 2014 - Honolulu, HI, United States
Duration: 29 Jun 20144 Jul 2014

Publication series

NameIEEE International Symposium on Information Theory - Proceedings
ISSN (Print)2157-8095


Conference2014 IEEE International Symposium on Information Theory, ISIT 2014
Country/TerritoryUnited States
CityHonolulu, HI


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