### Abstract

We consider the problem of recovering an N-dimensional sparse vector x from its linear transformation y=Dx of M (<N) dimensions. Minimization of the l_{1}-norm of x under the constraint y=Dx is a standard approach for the recovery problem, and earlier studies report that the critical condition for typically successful l_{1}-recovery is universal over a variety of randomly constructed matrices D. To examine the extent of the universality, we focus on the case in which D is provided by concatenating T=N/M matrices O _{1},O_{2},...,O_{T} drawn uniformly according to the Haar measure on the M×M orthogonal matrices. By using the replica method in conjunction with the development of an integral formula to handle the random orthogonal matrices, we show that the concatenated matrices can result in better recovery performance than that predicted by the universality when the density of non-zero signals is not uniform among the T matrix modules. The universal condition is reproduced for the special case of uniform non-zero signal densities. Extensive numerical experiments support the theoretical predictions.

Original language | English |
---|---|

Article number | P12003 |

Journal | Journal of Statistical Mechanics: Theory and Experiment |

Volume | 2012 |

Issue number | 12 |

DOIs | |

Publication status | Published - 1 Dec 2012 |

MoE publication type | A1 Journal article-refereed |

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### Keywords

- error correcting codes
- source and channel coding
- statistical inference

### Cite this

_{1}-recovery limit of sparse vectors represented by concatenations of random orthogonal matrices.

*Journal of Statistical Mechanics: Theory and Experiment*,

*2012*(12), [P12003]. https://doi.org/10.1088/1742-5468/2012/12/P12003

}

_{1}-recovery limit of sparse vectors represented by concatenations of random orthogonal matrices',

*Journal of Statistical Mechanics: Theory and Experiment*, vol. 2012, no. 12, P12003. https://doi.org/10.1088/1742-5468/2012/12/P12003

**Typical l _{1}-recovery limit of sparse vectors represented by concatenations of random orthogonal matrices.** / Kabashima, Yoshiyuki; Vehkaperä, Mikko; Chatterjee, Saikat.

Research output: Contribution to journal › Article › Scientific › peer-review

TY - JOUR

T1 - Typical l1-recovery limit of sparse vectors represented by concatenations of random orthogonal matrices

AU - Kabashima, Yoshiyuki

AU - Vehkaperä, Mikko

AU - Chatterjee, Saikat

PY - 2012/12/1

Y1 - 2012/12/1

N2 - We consider the problem of recovering an N-dimensional sparse vector x from its linear transformation y=Dx of M (<N) dimensions. Minimization of the l1-norm of x under the constraint y=Dx is a standard approach for the recovery problem, and earlier studies report that the critical condition for typically successful l1-recovery is universal over a variety of randomly constructed matrices D. To examine the extent of the universality, we focus on the case in which D is provided by concatenating T=N/M matrices O 1,O2,...,OT drawn uniformly according to the Haar measure on the M×M orthogonal matrices. By using the replica method in conjunction with the development of an integral formula to handle the random orthogonal matrices, we show that the concatenated matrices can result in better recovery performance than that predicted by the universality when the density of non-zero signals is not uniform among the T matrix modules. The universal condition is reproduced for the special case of uniform non-zero signal densities. Extensive numerical experiments support the theoretical predictions.

AB - We consider the problem of recovering an N-dimensional sparse vector x from its linear transformation y=Dx of M (<N) dimensions. Minimization of the l1-norm of x under the constraint y=Dx is a standard approach for the recovery problem, and earlier studies report that the critical condition for typically successful l1-recovery is universal over a variety of randomly constructed matrices D. To examine the extent of the universality, we focus on the case in which D is provided by concatenating T=N/M matrices O 1,O2,...,OT drawn uniformly according to the Haar measure on the M×M orthogonal matrices. By using the replica method in conjunction with the development of an integral formula to handle the random orthogonal matrices, we show that the concatenated matrices can result in better recovery performance than that predicted by the universality when the density of non-zero signals is not uniform among the T matrix modules. The universal condition is reproduced for the special case of uniform non-zero signal densities. Extensive numerical experiments support the theoretical predictions.

KW - error correcting codes

KW - source and channel coding

KW - statistical inference

UR - http://www.scopus.com/inward/record.url?scp=84871207413&partnerID=8YFLogxK

U2 - 10.1088/1742-5468/2012/12/P12003

DO - 10.1088/1742-5468/2012/12/P12003

M3 - Article

AN - SCOPUS:84871207413

VL - 2012

JO - Journal of Statistical Mechanics: Theory and Experiment

JF - Journal of Statistical Mechanics: Theory and Experiment

SN - 1742-5468

IS - 12

M1 - P12003

ER -

_{1}-recovery limit of sparse vectors represented by concatenations of random orthogonal matrices. Journal of Statistical Mechanics: Theory and Experiment. 2012 Dec 1;2012(12). P12003. https://doi.org/10.1088/1742-5468/2012/12/P12003