Analysis of sparse representations using bi-orthogonal dictionaries

Mikko Vehkaperä; Yoshiyuki Kabashima; Saikat Chatterjee; Erik Aurell; Mikael Skoglund; Lars Rasmussen

doi:10.1109/ITW.2012.6404757

Analysis of sparse representations using bi-orthogonal dictionaries

Mikko Vehkaperä, Yoshiyuki Kabashima, Saikat Chatterjee, Erik Aurell, Mikael Skoglund, Lars Rasmussen

Not published at VTT

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

2 Citations (Scopus)

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 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.

Original language	English
Title of host publication	2012 IEEE Information Theory Workshop, ITW 2012
Pages	647-651
ISBN (Electronic)	978-1-4673-0223-4
DOIs	https://doi.org/10.1109/ITW.2012.6404757
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 Sept 2012 → 7 Sept 2012

Conference

Conference	2012 IEEE Information Theory Workshop, ITW 2012
Country/Territory	Switzerland
City	Lausanne
Period	3/09/12 → 7/09/12

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10.1109/ITW.2012.6404757

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title = "Analysis of sparse representations using bi-orthogonal dictionaries",

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 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.",

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AU - Kabashima, Yoshiyuki

AU - Chatterjee, Saikat

AU - Aurell, Erik

AU - Skoglund, Mikael

AU - Rasmussen, Lars

PY - 2012/12/1

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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.

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Analysis of sparse representations using bi-orthogonal dictionaries

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