GRK 2088: Discovering structure in complex data: Statistics meets Optimization and Inverse Problems

从复杂数据中有效提取重要信息是应用科学的重大挑战之一。我们在RTG 2088中的研究工作旨在开发新的数学概念，以有效识别，提取和利用数据集中的相关结构信息。该RTG的指导原则之一是发现和严格利用数据中的结构先验信息。我们研究了广泛的先验知识-如拓扑结构和度量结构，形状空间形态学，稀疏性自适应字典或几何曲率泛函-设计数字和统计稳定和有效的算法，用于信息的提取和分类。在方法论上，我们专注于统计，优化和逆问题的方法之间的相互作用。重要的解决方案概念包括广义正则化技术，调和分析和统计中的多尺度方法，拓扑结构的统计推断，非线性局部和全局谱降维，以及随机和优化接口的迭代算法。在第一个资助期内，博士项目之间出现了几个新的重要联系。我们在第二个资助期的目标是深化和利用这些协同作用，以获得新的数据结构分析和利用方法，大大改善现有的方法。新的主题包括例如随机函数迭代的平均算子理论，分析非平稳信号的新参数方法，或在非参数正则化理论中使用浓度不等式。大多数拟议的博士项目研究和进一步发展数学方法，直接应用于生物学，医学或工程学。RTG开发了一个跨学科的资格课程，为我们的研究生提供了一套广泛的技能和多功能工具，用于分析RTG科学范围之外的数据。这为学术界或工业界的成功职业生涯提供了有针对性的准备。博士课程“数学科学”在哥廷根大学，特别强调国际科学交流，这是通过面向研究和教育的访问者计划，并通过支持研究生在国外的长期研究访问来实现的。

The efficient extraction of significant information from complex data is one of the grand challenges in applied sciences. Our research work in the RTG 2088 aims at developing new mathematical concepts for efficient identification, extraction and utilization of relevant structural information in data sets. One of the guiding principles of this RTG consists of discovering and rigorously exploiting structural a priori information in data. We study a wide range of a priori knowledge – such as topological and metric structures, shape space morphology, sparsity in adaptive dictionaries or geometric curvature functionals – to design numerically and statistically stable and efficient algorithms for the extraction and classification of information. Methodologically, we focus on an interplay between approaches in statistics, optimization, and inverse problems. Important solution concepts include generalized regularization techniques, multi-scale methods in harmonic analysis and statistics, statistical inference for topological structures, nonlinear local and global spectral dimensionality reduction, and iterative algorithms at the interface of stochastics and optimization. Within the first funding period, several new important connections between the PhD projects have emerged. Our goal in the second funding period is to deepen and exploit these synergies in order to derive new approaches for data structure analysis and utilization that significantly improve the existing methods. New topics involve for example the theory of averaging operators for random function iterations, new parametric methods to analyze non-stationary signals, or the use of concentration inequalities in nonparametric regularization theory. Most of the proposed PhD projects investigate and further develop mathematical methods with a direct application in biology, medicine or engineering. The RTG has developed an interdisciplinary qualification program which equips our graduate students with a broad set of skills and versatile tools for analyzing data also beyond the scientific scope of the RTG. This provides targeted preparation for successful careers both in academia or industry. The PhD program “Mathematical Sciences” at the University of Göttingen, puts special emphasis on international scientific exchange which is achieved by a visitor program geared towards research and education and by the support of extended research visits abroad for graduate students.