Preasymptotic error analysis for function recovery problems in high dimensions

点击翻译按钮获取中文摘要

Many applications in engineering, science, and statistics require inter- or extrapolation from data. Mathematically speaking, the problem is to find a function fitting the data. This research project is concerned with the preasymptotic error analysis of such recoverry problems for high-dimensional data. The functions appearing in the studied recovery problems are subject to two different kinds of model assumptions: on the one hand, the boundedness of mixed derivatives and generalizations thereof, which naturally appear in the context of the electronic Schrödinger equation and sparse grid methods; on the other hand, assumptions of structured dependencies, which are significant in semiparametric statistics and machine learning.The main focus of the research project are preasymptotic bounds for worst-case errors and the design of optimal algorithms given one of the previously mentioned model assumptions. Worst-case error estimates are a central ingredient in the analysis of approximation and function recovery methods. They provide a priori error estimates which are most reliable given correct model assumptions. At the same time, worst-case error estimates give insights into the fundamental limitations of approximation and recovery methods.For the considered problems, asymptotic error estimates are typically known for quite some time. In case of functions defined on high-dimensional domains, however, asymptotic estimates often turn out to be useless. One reason is that they are only valid after investing a number of samples growing exponentially with the dimension. At this point, preasymptotics become crucial to obtain practically relevant error estimates. Preasymptotics are also important to precisely determine the level of tractability of a high-dimensional approximation problem. In particular, preasymptotics allow to decide whether or not the curse of dimensionality is present.The rigorous analysis of preasymptotics in this research project will be based on fundamental concepts and results from approximation theory and functional analysis. The concept particularly worth mentioning is metric entropy. Metric entropy, respectively entropy numbers, is an essential ingredient in the context of Carl's inequality, concentration inequalities for empirical processes, and a new characterization method for worst-case errors of Sobolev embeddings discovered by the applicant and coauthors. All three will be important tools in the proofs of lower and upper bounds for worst-case errors.