Preasymptotic error analysis for function recovery problems in high dimensions

工程、科学和统计中的许多应用都需要从数据中进行内推或外推。从数学上讲，问题是找到一个符合数据的函数。本文主要研究高维数据恢复问题的预渐近误差分析。所研究的恢复问题中出现的函数受制于两种不同的模型假设：一方面，混合导数及其推广的有界性，这自然地出现在电子薛定谔方程和稀疏网格方法的背景下；另一方面，结构依赖的假设，这在半参数统计和机器学习中是重要的。研究项目的主要焦点是最坏情况误差的预渐近界和给定上述模型假设之一的最优算法的设计。最坏情况误差估计是分析近似和函数恢复方法的核心内容。它们提供了一个先验误差估计，在正确的模型假设下是最可靠的。同时，最坏情况误差估计揭示了近似和恢复方法的基本局限性。对于所考虑的问题，通常已知渐近误差估计相当长一段时间。然而，对于定义在高维区域上的函数，渐近估计通常被证明是无用的。一个原因是，它们只有在投资了大量随着维度呈指数增长的样本后才有效。在这一点上，预渐近性成为获得实际相关误差估计的关键。预符号学对于精确确定高维近似问题的可处理程度也很重要。特别是，预渐近性可以决定是否存在维度灾难。在本研究项目中，对预渐近性的严格分析将基于逼近理论和泛函分析的基本概念和结果。特别值得一提的是度量熵。度量熵，分别是熵数，是卡尔不等式、经验过程的浓度不等式以及申请人和合著者发现的Sobolev嵌入的最坏情况错误的一种新的刻画方法。在证明最坏情况下的错误的下界和上界时，这三个都将是重要的工具。

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.