Separation-Free Spectral Super-Resolution via Convex Optimization

TL;DR

This paper introduces a weighting scheme for atomic norm methods that achieves arbitrarily high resolution in spectral super-resolution without separation constraints, even in the absence of noise.

Contribution

It proposes a simple weighting scheme that enables separation-free super-resolution with convex optimization, overcoming resolution limitations of existing atomic norm methods.

Findings

Achieves arbitrarily high resolution in noise-free scenarios.

Provides a kernel-free dual certificate construction.

Numerical results validate theoretical claims.

Abstract

Atomic norm methods have recently been proposed for spectral super-resolution with flexibility in dealing with missing data and miscellaneous noises. A notorious drawback of these convex optimization methods however is their lower resolution in the high signal-to-noise (SNR) regime as compared to conventional methods such as ESPRIT. In this paper, we devise a simple weighting scheme in existing atomic norm methods and show that the resolution of the resulting convex optimization method can be made arbitrarily high in the absence of noise, achieving the so-called separation-free super-resolution. This is proved by a novel, kernel-free construction of the dual certificate whose existence guarantees exact super-resolution using the proposed method. Numerical results corroborating our analysis are provided.