Super-Resolution Meets Machine Learning: Approximation of Measures

Abstract

The problem of super-resolution in general terms is to recuperate a finitely supported measure µ given finitely many of its coefficientsμ(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 µ.In this paper, we consider the more severe problem of recuperating µ approximately without any assumption on µ beyond having a finite total variation. In particular, µ may be supported on a continuum, so that the minimal separation among the points in the support of µ 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 µ 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. This paper is motivated by two apparently disjoint areas; super-resolution and machine learning. A problem of interest in both of these areas is the approximation of a measure using a finite amount of information on the measure. Thus we wish to develop a theory of (weak-star) approximation of measures. We will describe our motivation and the connections of this work to the problem of super-resolution and the problem of machine learning in Sections 1.1 and 1.2 respectively. The aims and contributions of this paper, and its outline is given in Section 1.3. This section and the next being introductory in nature, the notation used in these two sections may not be the same as the one used in the remainder of the paper. Super-resolutionThe problem of super-resolution is stated by Donoho [12] as follows. Given observations of the form k∈Z a k exp(−iωk∆) + z(ω), |ω| ≤ Ω, (1.1)where {a k } is sequence of complex numbers, ∆, Ω > 0, and z represents a perturbation subject to the condition that Ω −Ω |z(ω)| 2 dω ≤ ǫ, recuperate the sequence {a k } up to an accuracy O(ǫ) in the sense of optimal recovery, when Ω∆ < π. It is shown in [12] that this is not possible in general, but it is possible under some sparsity assumptions.Relevant to the current paper is a generalization stated by Candés and Fernandez-Granda in [6, Theorem 1.2]. We denote the quotient space R/(2πZ) by T. Let µ(t) = K k=1 a k δ(t − t k ) + Z(t),(1.2)where δ denotes the Dirac delta measure at 0. We assume that the moments K k=1 a k exp(−ijt k ) + T Z(t) exp(−ijt)dt,

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