Uniform Uncertainty Principle and Signal Recovery via Regularized Orthogonal Matching Pursuit

Authors:
Deanna Needell;Roman Vershynin
Affiliations:
University of California, Department of Mathematics, One Shields Ave, 95616, Davis, CA, USA;University of California, Department of Mathematics, One Shields Ave, 95616, Davis, CA, USA
Venue:
Foundations of Computational Mathematics
Year:
2009

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Abstract

This paper seeks to bridge the two major algorithmic approaches to sparse signal recovery from an incomplete set of linear measurements—L1-minimization methods and iterative methods (Matching Pursuits). We find a simple regularized version of Orthogonal Matching Pursuit (ROMP) which has advantages of both approaches: the speed and transparency of OMP and the strong uniform guarantees of L1-minimization. Our algorithm, ROMP, reconstructs a sparse signal in a number of iterations linear in the sparsity, and the reconstruction is exact provided the linear measurements satisfy the uniform uncertainty principle.