### Abstract

We consider the estimation of a random vector observed through a linear transform followed by a componentwise probabilistic measurement channel. Although such linear mixing estimation problems are generally highly non-convex, Gaussian approximations of belief propagation (BP) have proven to be computationally attractive and highly effective in a range of applications. Recently, Bayati and Montanari have provided a rigorous and extremely general analysis of a large class of approximate message passing (AMP) algorithms that includes many Gaussian approximate BP methods. This paper extends their analysis to a larger class of algorithms to include what we call generalized AMP (G-AMP). G-AMP incorporates general (possibly non-AWGN) measurement channels. Similar to the AWGN output channel case, we show that the asymptotic behavior of the G-AMP algorithm under large i.i.d. Gaussian transform matrices is described by a simple set of state evolution (SE) equations. The general SE equations recover and extend several earlier results, including SE equations for approximate BP on general output channels by Guo and Wang.

Original language | English (US) |
---|---|

Title of host publication | 2011 IEEE International Symposium on Information Theory Proceedings, ISIT 2011 |

Pages | 2168-2172 |

Number of pages | 5 |

DOIs | |

State | Published - 2011 |

Event | 2011 IEEE International Symposium on Information Theory Proceedings, ISIT 2011 - St. Petersburg, Russian Federation Duration: Jul 31 2011 → Aug 5 2011 |

### Other

Other | 2011 IEEE International Symposium on Information Theory Proceedings, ISIT 2011 |
---|---|

Country | Russian Federation |

City | St. Petersburg |

Period | 7/31/11 → 8/5/11 |

### Fingerprint

### Keywords

- belief propagation
- compressed sensing
- estimation
- Optimization
- random matrices

### ASJC Scopus subject areas

- Applied Mathematics
- Modeling and Simulation
- Theoretical Computer Science
- Information Systems

### Cite this

*2011 IEEE International Symposium on Information Theory Proceedings, ISIT 2011*(pp. 2168-2172). [6033942] https://doi.org/10.1109/ISIT.2011.6033942

**Generalized approximate message passing for estimation with random linear mixing.** / Rangan, Sundeep.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*2011 IEEE International Symposium on Information Theory Proceedings, ISIT 2011.*, 6033942, pp. 2168-2172, 2011 IEEE International Symposium on Information Theory Proceedings, ISIT 2011, St. Petersburg, Russian Federation, 7/31/11. https://doi.org/10.1109/ISIT.2011.6033942

}

TY - GEN

T1 - Generalized approximate message passing for estimation with random linear mixing

AU - Rangan, Sundeep

PY - 2011

Y1 - 2011

N2 - We consider the estimation of a random vector observed through a linear transform followed by a componentwise probabilistic measurement channel. Although such linear mixing estimation problems are generally highly non-convex, Gaussian approximations of belief propagation (BP) have proven to be computationally attractive and highly effective in a range of applications. Recently, Bayati and Montanari have provided a rigorous and extremely general analysis of a large class of approximate message passing (AMP) algorithms that includes many Gaussian approximate BP methods. This paper extends their analysis to a larger class of algorithms to include what we call generalized AMP (G-AMP). G-AMP incorporates general (possibly non-AWGN) measurement channels. Similar to the AWGN output channel case, we show that the asymptotic behavior of the G-AMP algorithm under large i.i.d. Gaussian transform matrices is described by a simple set of state evolution (SE) equations. The general SE equations recover and extend several earlier results, including SE equations for approximate BP on general output channels by Guo and Wang.

AB - We consider the estimation of a random vector observed through a linear transform followed by a componentwise probabilistic measurement channel. Although such linear mixing estimation problems are generally highly non-convex, Gaussian approximations of belief propagation (BP) have proven to be computationally attractive and highly effective in a range of applications. Recently, Bayati and Montanari have provided a rigorous and extremely general analysis of a large class of approximate message passing (AMP) algorithms that includes many Gaussian approximate BP methods. This paper extends their analysis to a larger class of algorithms to include what we call generalized AMP (G-AMP). G-AMP incorporates general (possibly non-AWGN) measurement channels. Similar to the AWGN output channel case, we show that the asymptotic behavior of the G-AMP algorithm under large i.i.d. Gaussian transform matrices is described by a simple set of state evolution (SE) equations. The general SE equations recover and extend several earlier results, including SE equations for approximate BP on general output channels by Guo and Wang.

KW - belief propagation

KW - compressed sensing

KW - estimation

KW - Optimization

KW - random matrices

UR - http://www.scopus.com/inward/record.url?scp=80054799706&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80054799706&partnerID=8YFLogxK

U2 - 10.1109/ISIT.2011.6033942

DO - 10.1109/ISIT.2011.6033942

M3 - Conference contribution

SN - 9781457705953

SP - 2168

EP - 2172

BT - 2011 IEEE International Symposium on Information Theory Proceedings, ISIT 2011

ER -