Super-Resolution Meets Machine Learning: Approximation of Measures

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.