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,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.
Keywords
Cite
@article{arxiv.1204.4065,
title = {Analysis of Sparse Representations Using Bi-Orthogonal Dictionaries},
author = {Mikko Vehkaperä and Yoshiyuki Kabashima and Saikat Chatterjee and Erik Aurell and Mikael Skoglund and Lars Rasmussen},
journal= {arXiv preprint arXiv:1204.4065},
year = {2013}
}
Comments
5 pages, 2 figures. The main result and numerical examples have been revised