Analysis of Sparse Representations Using Bi-Orthogonal Dictionaries

TL;DR

This paper analyzes the performance of l1-reconstruction for sparse signals using bi-orthogonal dictionaries, deriving conditions for perfect recovery through the replica method.

Contribution

It introduces a novel analysis of sparse recovery using bi-orthogonal dictionaries, extending prior work on IID Gaussian dictionaries.

Findings

Identifies critical conditions for perfect l1-recovery with bi-orthogonal dictionaries

Uses replica method to analyze sparse recovery performance

Provides theoretical insights into structured dictionary design

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.