Preasymptotic error analysis for function recovery problems in high dimensions

在工程、科学和统计学中的许多应用需要从数据中进行内推或外推。从数学上讲，问题是找到一个拟合数据的函数。本研究计画系关于高维资料之复原问题之预渐近误差分析。在所研究的恢复问题中出现的函数受到两种不同类型的模型假设的影响：一方面，混合导数及其推广的有界性，这自然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，结构化依赖的假设，在半参数统计和机器学习中具有重要意义。本研究项目的主要重点是最坏估计的预渐近界，案例误差和最优算法的设计给出了前面提到的模型假设之一。最坏情况下的误差估计是近似和函数恢复方法分析的核心内容。他们提供了一个先验的误差估计，这是最可靠的给定正确的模型假设。同时，最坏情况下的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，渐近误差估计通常在相当长的一段时间内是已知的。然而，在高维域上定义的函数的情况下，渐近估计往往是无用的。一个原因是，它们只有在投资了大量随维度呈指数增长的样本后才有效。在这一点上，预渐近性变得至关重要，以获得实际相关的误差估计。前征兆对于精确确定高维近似问题的易处理性水平也很重要。特别是，预渐近允许决定是否存在维数灾难。在这个研究项目中，预渐近的严格分析将基于近似理论和泛函分析的基本概念和结果。特别值得一提的概念是度量熵。度量熵，相应地熵数，是卡尔不等式、经验过程的浓度不等式以及由申请人和共同作者发现的索伯列夫嵌入的最坏情况错误的新表征方法的背景下的重要成分。所有这三个将是重要的工具，在证明下限和上限的最坏情况下的错误。

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.