Super-Resolution Meets Machine Learning: Approximation of Measures

• Published in: The Journal of fourier analysis and applications Vol. 25; no. 6; pp. 3104 - 3122
• Main Author: Mhaskar, H. N.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-12-2019; Springer; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Approximation; Approximations and Expansions; Artificial intelligence; Convolution; Fourier Analysis; Inverse problems; Machine learning; Mathematical analysis; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Partial Differential Equations; Separation; Signal,Image and Speech Processing; Widths; Super-resolution; 41A25; 68T05; Machine learning; Data defined spaces; 65J22; De-convolution; 94A12
• ISSN: 1069-5869; 1531-5851
• DOI: 10.1007/s00041-019-09693-x

The problem of super-resolution in general terms is to recuperate a finitely supported measure
μ
given finitely many of its coefficients
μ
^
(
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.