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arXiv · 1204.4065

Analysis of Sparse Representations Using Bi-Orthogonal Dictionaries

Mikko Vehkaperä·Yoshiyuki Kabashima·Saikat Chatterjee·Erik Aurell·Mikael Skoglund·Lars Rasmussen
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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,O2 are independent and drawn uniformly according to the Haar measure on the group of orthogonal M x M matrices. By an application of the replica method, we obtain the critical conditions under which perfect l1-recovery is possible with bi-orthogonal dictionaries.

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Mikko Vehkaperä, Yoshiyuki Kabashima, Saikat Chatterjee, Erik Aurell, Mikael Skoglund, Lars Rasmussen. 2012-04-18. Analysis of Sparse Representations Using Bi-Orthogonal Dictionaries. https://doi.org/10.1109/itw.2012.6404757

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