TY - GEN

T1 - Asymptotic MMSE analysis under sparse representation modeling

AU - Huleihel, Wasim

AU - Merhav, Neri

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84906568548&partnerID=8YFLogxK

U2 - 10.1109/ISIT.2014.6875311

DO - 10.1109/ISIT.2014.6875311

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AN - SCOPUS:84906568548

SN - 9781479951864

T3 - IEEE International Symposium on Information Theory - Proceedings

SP - 2634

EP - 2638

BT - 2014 IEEE International Symposium on Information Theory, ISIT 2014

PB - Institute of Electrical and Electronics Engineers Inc.

T2 - 2014 IEEE International Symposium on Information Theory, ISIT 2014

Y2 - 29 June 2014 through 4 July 2014

ER -