### 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 language | English (US) |
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Title of host publication | Advances in Neural Information Processing Systems 22 - Proceedings of the 2009 Conference |

Pages | 1545-1553 |

Number of pages | 9 |

State | Published - 2009 |

Event | 23rd Annual Conference on Neural Information Processing Systems, NIPS 2009 - Vancouver, BC, Canada Duration: Dec 7 2009 → Dec 10 2009 |

### Other

Other | 23rd Annual Conference on Neural Information Processing Systems, NIPS 2009 |
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Country | Canada |

City | Vancouver, BC |

Period | 12/7/09 → 12/10/09 |

### Fingerprint

### ASJC Scopus subject areas

- Information Systems

### Cite this

*Advances in Neural Information Processing Systems 22 - Proceedings of the 2009 Conference*(pp. 1545-1553)

**Asymptotic analysis of MAP estimation via the replica method and compressed sensing.** / Rangan, Sundeep; Fletcher, Alyson K.; Goyal, Vivek K.

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

*Advances in Neural Information Processing Systems 22 - Proceedings of the 2009 Conference.*pp. 1545-1553, 23rd Annual Conference on Neural Information Processing Systems, NIPS 2009, Vancouver, BC, Canada, 12/7/09.

}

TY - GEN

T1 - Asymptotic analysis of MAP estimation via the replica method and compressed sensing

AU - Rangan, Sundeep

AU - Fletcher, Alyson K.

AU - Goyal, Vivek K.

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84858722178&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84858722178&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84858722178

SN - 9781615679119

SP - 1545

EP - 1553

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

ER -