Thursday, September 04, 2014

Enhancing Sparsity and Resolution via Reweighted Atomic Norm Minimization

Yet another clear use of sharp phase transitions as an acid test for the purpose of delineating what algorithm does well and which one does ... less well. Outstanding ! Let us hope the implementation comes about at some point in time in the future:

Enhancing Sparsity and Resolution via Reweighted Atomic Norm Minimization by Zai YangLihua Xie

The mathematical theory of super-resolution developed recently by Cand\`{e}s and Fernandes-Granda states that a continuous, sparse frequency spectrum can be recovered with infinite precision via a (convex) atomic norm technique given a set of regularly spaced time-space samples. This theory was then extended to the cases with partial/compressive samples and/or multiple measurement vectors via atomic norm minimization (ANM), known as off-grid/continuous compressed sensing. However, a major problem of existing atomic norm methods is that the frequencies can be recovered only if they are sufficiently separated, prohibiting commonly known high resolution. In this paper, a novel nonconvex optimization method is proposed which guarantees exact recovery under no resolution limit and hence achieves high resolution. A locally convergent iterative algorithm is implemented to solve the nonconvex problem. The algorithm iteratively carries out ANM with a sound reweighting strategy which enhances sparsity and resolution, and is termed as reweighted atomic-norm minimization (RAM). Extensive numerical simulations are carried out to demonstrate the performance of the proposed method with application to direction of arrival (DOA) estimation.

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