### Abstract

The sparse representation problem of recovering an N dimensional sparse vector x from M < N linear observations y = Dx given dictionary D is considered. The standard approach is to let the elements of the dictionary be independent and identically distributed (IID) zero-mean Gaussian and minimize the l_{1}-norm of x under the constraint y = Dx. In this paper, the performance of l_{1}-reconstruction is analyzed, when the dictionary is bi-orthogonal D = [O_{1} O_{2}], where O_{1}, O _{2} are independent and drawn uniformly according to the Haar measure on the group of orthogonal M × M matrices. By an application of the replica method, we obtain the critical conditions under which perfect l _{1}-recovery is possible with bi-orthogonal dictionaries.

Original language | English |
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Title of host publication | 2012 IEEE Information Theory Workshop, ITW 2012 |

Pages | 647-651 |

ISBN (Electronic) | 978-1-4673-0223-4 |

DOIs | |

Publication status | Published - 1 Dec 2012 |

MoE publication type | A4 Article in a conference publication |

Event | 2012 IEEE Information Theory Workshop, ITW 2012 - Lausanne, Switzerland Duration: 3 Sep 2012 → 7 Sep 2012 |

### Conference

Conference | 2012 IEEE Information Theory Workshop, ITW 2012 |
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Country | Switzerland |

City | Lausanne |

Period | 3/09/12 → 7/09/12 |

### Fingerprint

### Cite this

*2012 IEEE Information Theory Workshop, ITW 2012*(pp. 647-651). [6404757] https://doi.org/10.1109/ITW.2012.6404757

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*2012 IEEE Information Theory Workshop, ITW 2012.*, 6404757, pp. 647-651, 2012 IEEE Information Theory Workshop, ITW 2012, Lausanne, Switzerland, 3/09/12. https://doi.org/10.1109/ITW.2012.6404757

**Analysis of sparse representations using bi-orthogonal dictionaries.** / Vehkaperä, Mikko; Kabashima, Yoshiyuki; Chatterjee, Saikat; Aurell, Erik; Skoglund, Mikael; Rasmussen, Lars.

Research output: Chapter in Book/Report/Conference proceeding › Conference article in proceedings › Scientific › peer-review

TY - GEN

T1 - Analysis of sparse representations using bi-orthogonal dictionaries

AU - Vehkaperä, Mikko

AU - Kabashima, Yoshiyuki

AU - Chatterjee, Saikat

AU - Aurell, Erik

AU - Skoglund, Mikael

AU - Rasmussen, Lars

PY - 2012/12/1

Y1 - 2012/12/1

N2 - The sparse representation problem of recovering an N dimensional sparse vector x from M < N linear observations y = Dx given dictionary D is considered. The standard approach is to let the elements of the dictionary be independent and identically distributed (IID) zero-mean Gaussian and minimize the l1-norm of x under the constraint y = Dx. In this paper, the performance of l1-reconstruction is analyzed, when the dictionary is bi-orthogonal D = [O1 O2], where O1, O 2 are independent and drawn uniformly according to the Haar measure on the group of orthogonal M × M matrices. By an application of the replica method, we obtain the critical conditions under which perfect l 1-recovery is possible with bi-orthogonal dictionaries.

AB - The sparse representation problem of recovering an N dimensional sparse vector x from M < N linear observations y = Dx given dictionary D is considered. The standard approach is to let the elements of the dictionary be independent and identically distributed (IID) zero-mean Gaussian and minimize the l1-norm of x under the constraint y = Dx. In this paper, the performance of l1-reconstruction is analyzed, when the dictionary is bi-orthogonal D = [O1 O2], where O1, O 2 are independent and drawn uniformly according to the Haar measure on the group of orthogonal M × M matrices. By an application of the replica method, we obtain the critical conditions under which perfect l 1-recovery is possible with bi-orthogonal dictionaries.

UR - http://www.scopus.com/inward/record.url?scp=84873181807&partnerID=8YFLogxK

U2 - 10.1109/ITW.2012.6404757

DO - 10.1109/ITW.2012.6404757

M3 - Conference article in proceedings

AN - SCOPUS:84873181807

SN - 978-1-4673-0224-1

SP - 647

EP - 651

BT - 2012 IEEE Information Theory Workshop, ITW 2012

ER -