# Super-resolution meets machine learning: approximation of measures

## TL;DR

This paper investigates the problem of approximately recovering measures from limited information, extending super-resolution concepts to measures supported on continua, with explicit recovery operators and optimal error estimates.

## Contribution

It introduces a new framework for measure approximation without support separation assumptions, providing explicit recovery operators and optimal error bounds.

## Key findings

- Explicit recovery operator for measures
- Optimal bounds on approximation error
- Recovery limitations for limited information

## Abstract

The problem of super-resolution in general terms is to recuperate a finitely supported measure $\mu$ given finitely many of its coefficients $\hat{\mu}(k)$ with respect to some orthonormal system. The interesting case concerns situations, where the number of coefficients required is substantially smaller than a power of the reciprocal of the minimal separation among the points in the support of $\mu$. In this paper, we consider the more severe problem of recuperating $\mu$ approximately without any assumption on $\mu$ beyond having a finite total variation. In particular, $\mu$ may be supported on a continuum, so that the minimal separation among the points in the support of $\mu$ is $0$. A variant of this problem is also of interest in machine learning as well as the inverse problem of de-convolution. We define an appropriate notion of a distance between the target measure and its recuperated version, give an explicit expression for the recuperation operator, and estimate the distance between $\mu$ and its approximation. We show that these estimates are the best possible in many different ways. We also explain why for a finitely supported measure the approximation quality of its recuperation is bounded from below if the amount of information is smaller than what is demanded in the super-resolution problem.