Coherence-based performance guarantees for estimating a sparse vector under random noise

Authors:
Zvika Ben-Haim;Yonina C. Eldar;Michael Elad
Affiliations:
Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa, Israel;Stanford University, Stanford, CA;Department of Computer Science, Technion-Israel Institute of Technology, Haifa, Israel
Venue:
IEEE Transactions on Signal Processing
Year:
2010

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Abstract

We consider the problem of estimating a deterministic sparse vector X0 from underdetermined measurements AX0 + w, where w represents white Gaussian noise and A is a given deterministic dictionary. We provide theoretical performance guarantees for three sparse estimation algorithms: basis pursuit denoising (BPDN), orthogonal matching pursuit (OMP), and thresholding. The performance of these techniques is quantified as the l2 distance between the estimate and the true value of X0. We demonstrate that, with high probability, the analyzed algorithms come close to the behavior of the oracle estimator, which knows the locations of the nonzero elements in X0. Our results are non-asymptotic and are based only on the coherence of A, so that they are applicable to arbitrary dictionaries. This provides insight on the advantages and drawbacks of l1 relaxation techniques such as BPDN and the Dantzig selector, as opposed to greedy approaches such as OMP and thresholding.