Discrepancy and universal discretization of frames

The central goal of this project is the development of mathematical theory that enables the construction of novel, universal reduction schemes for highly redundant dictionaries of functions. The mathematical terminology continuous frames is used to describe such dictionaries, which can be used to decompose complex functions (or data) into simple building blocks, so-called atoms. Such decompositions are used to illuminate the structure and properties of functions (or data) under scrutiny. However, continuous frames often consist of uncountably many atoms, i.e., an infinite number that cannot even be enumerated. Hence, the reduction of the dictionary size is essential for practical use of continuous frames. Ideally, desirable properties of the original dictionary are retained as faithfully as possible. Although theoretical results suggest that such reduction schemes exist for almost any continuous frame, prior constructions usually leverage known structure of a given dictionary to select, which atoms are selected for the reduced dictionary. Therefore, they do not easily translate to other dictionaries, forming a fundamental obstacle for their usage, practical and theoretical. The lack of universal reduction schemes is an inhibitor to realizing the full potential of continuous frames for data analysis and provides the core motivation for the research proposed in this project: We investigate reduction schemes that can be applied under mild and rather general conditions on the dictionary, i.e., they are universal. This step is crucial to facilitate the use of more general dictionaries. To this end, we use so-called low discrepancy sets, a number-theoretic construction through which we can guarantee that the methods developed in this project are applicable to a large variety of continuous frames. This is a completely novel application of low discrepancy sets, the most successful application of which has been the computational approximation of high-dimensional integrals by means of point evaluations.