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.
|Title of host publication||2012 IEEE Information Theory Workshop, ITW 2012|
|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||2012 IEEE Information Theory Workshop, ITW 2012|
|Period||3/09/12 → 7/09/12|