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

GRK 2088：发现复杂数据中的结构：统计遇到优化和反问题

• 批准号: 257660731
• 金额: --
• 依托单位: Georg-August-Universität Göttingen
• 依托单位国家: 德国
• 项目类别: Research Training Groups
• 财政年份: 2015
• 资助国家: 德国
• 起止时间: 2014-12-31 至 2023-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/257660731?language=en
• 关键词: GRK; 2088; Discovering; structure; complex

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.

从复杂数据中高效地提取有意义的信息是应用科学面临的重大挑战之一。我们在RTG 2088中的研究工作旨在开发新的数学概念，以有效地识别、提取和利用数据集中的相关结构信息。这种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，如拓扑和度量结构、形状空间形态、自适应词典中的稀疏性或几何曲率泛函，以设计数值和统计稳定的高效算法来提取和分类信息。在方法上，我们专注于统计、优化和反问题方法之间的相互作用。重要的解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，以及随机和优化交界处的迭代算法。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，以便得出新的数据结构分析和利用方法，大大改进现有方法。例如，新的主题涉及随机函数迭代的平均算子理论，分析非平稳信号的新的参数方法，或非参数正则化理论中浓度不等式的使用。大多数被提议的博士项目调查并进一步发展数学方法，直接应用于生物学、医学或工程学。RTG开发了一个跨学科资格认证计划，使我们的研究生具备广泛的技能和通用工具来分析也超出RTG科学范围的数据。这为在学术界或行业取得成功的职业生涯提供了有针对性的准备。哥廷根大学的博士项目“数学科学”特别强调国际科学交流，这是通过面向研究和教育的访客项目以及对研究生延长海外研究访问的支持来实现的。