%0 Journal Article
%T Analysis of Sparse Representations Using Bi-Orthogonal Dictionaries
%A Mikko Vehkaper£¿
%A Yoshiyuki Kabashima
%A Saikat Chatterjee
%A Erik Aurell
%A Mikael Skoglund
%A Lars Rasmussen
%J Mathematics
%D 2012
%I arXiv
%R 10.1109/ITW.2012.6404757
%X 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.
%U http://arxiv.org/abs/1204.4065v2