Approximate super-resolution of positive measures in all dimensions

TL;DR

This paper introduces new methods and theoretical results for reconstructing positive measures from approximate moments using convex optimization, with applications to super-resolution in multiple dimensions.

Contribution

It provides new uniqueness theorems, quantitative recovery estimates, and a sum-of-squares hierarchy for approximate super-resolution on semi-algebraic sets.

Findings

New uniqueness results for measure reconstruction

Quantitative estimates for approximate recovery

A sum-of-squares hierarchy for super-resolution

Abstract

We study the problem of reconstructing a positive discrete measure on a compact set $K \subseteq \mathbb{R}^n$ from a finite set of moments (possibly known only approximately) via convex optimization. We give new uniqueness results, new quantitative estimates for approximate recovery and a new sum-of-squares based hierarchy for approximate super-resolution on compact semi-algebraic sets.