Analysis of Sparse Representations Using Bi-Orthogonal Dictionaries

Details Analysis of Sparse Representations Using Bi-Orthogonal Dictionaries Analysis of Sparse Representations Using Bi-Orthogonal Dictionaries

2012-04-18

arxiv.org/abs/1204.4065v1

Computer Science

Information Theory

5 pages, 1 figure

Scientific paper

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 show that the typical compression threshold for such D is the same as for the IID Gaussian dictionary.

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