Coherence-Based Performance Guarantees for Estimating a Sparse Vector Under Random Noise

We consider the problem of estimating a deterministic sparse vector x(0) from underdetermined measurements Ax(0) + 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 l(2) distance between the estimate and the true value of x(0). 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 x(0). 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 l(1) relaxation techniques such as BPDN and the Dantzig selector, as opposed to greedy approaches such as OMP and thresholding.