A Tight Converse to the Spectral Resolution Limit via Convex Programming

TL;DR

This paper proves the necessity of a conjectured phase transition in spectral component estimation using convex programming, showing failure when spectral components are too close, specifically at a distance of 1/m.

Contribution

It provides a rigorous proof of the necessity part of a conjecture regarding the spectral resolution limit in convex spectral estimation methods.

Findings

Convex programming can fail when spectral components are at a distance of 1/m.

The paper establishes the sharpness of the spectral resolution limit.

It confirms the conjectured phase transition in spectral estimation success.

Abstract

It is now well understood that convex programming can be used to estimate the frequency components of a spectrally sparse signal from $2m+1$ uniform temporal measurements. It is conjectured that a phase transition on the success of the total-variation regularization occurs when the distance between the spectral components of the signal to estimate crosses $1/m$. We prove the necessity part of this conjecture by demonstrating that this regularization can fail whenever the spectral distance of the signal of interest is asymptotically equal to $1/m$.