Enhancing Sparsity and Resolution via Reweighted Atomic Norm Minimization

TL;DR

This paper introduces reweighted atomic-norm minimization (RAM), a novel iterative method that enhances sparsity and resolution in super-resolution problems, outperforming traditional atomic norm techniques especially when frequencies are closely spaced.

Contribution

The paper proposes a nonconvex sparse metric and a RAM algorithm that iteratively improves super-resolution recovery beyond existing atomic norm methods.

Findings

RAM achieves higher resolution than standard atomic norm minimization.

Numerical simulations show RAM's superior performance in DOA estimation.

RAM effectively recovers closely spaced frequencies with improved accuracy.

Abstract

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 uniform time-space samples. This theory was then extended to the cases of partial/compressive samples and/or multiple measurement vectors via atomic norm minimization (ANM), known as off-grid/continuous compressed sensing (CCS). 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) sparse metric is proposed that promotes sparsity to a greater extent than the atomic norm. Using this metric an optimization problem is formulated and a locally convergent iterative algorithm is implemented. 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 advantageous performance of RAM with application to direction of arrival (DOA) estimation.