AF: Small: Bundle-theoretic methods for local-to-global inference

现代技术，如蜂窝设备和其他复杂的传感器，使收集大量数据成为可能。在许多应用程序中，结果数据可能来自不同的来源和不同的时间，或者信息量如此之大，以至于不可能在一台计算机中进行分析或存储。此类应用程序的示例包括云中的分析，或者出于隐私和/或安全原因从分布在设备网络中的数据库进行推断。分析这类数据的需求促使数学、统计和计算方面的挑战不断增加，需要将分布式测量集合起来得出关于系统的结论。迄今为止，大多数研究都集中在高效、准确地同步/对齐分布式数据的计算问题上，但很少有人知道如何解决找到这种解决方案的数学障碍。这就是本项目试图解决的知识鸿沟。具体来说，研究者将发展从分布式数据中学习所需的理论、数学和算法基础，即使在不可能完全同步的情况下。该项目开发的工具将推动纯数学和计算数学的发展，并有可能应用于云计算、分布式数据可视化和传感器融合等领域。这项研究的影响将进一步扩大，因为它被纳入研究者开发的新的教育材料，以及在计算机科学家和数学家的培训中。该项目的主要主题是将经典代数拓扑（数学的分支，关注抽象对象的形状和局部结构如何相互作用）中的工具应用于从分布式数据中学习的问题。具体来说，由该奖项资助的研究旨在：(1)利用束理论的思想，以开发能够从数据中估计拓扑障碍到分布式测量的全球同步的算法；并且(2)利用光纤束理论的工具来计算本地数据的一致集合，即使在存在非平凡障碍物的情况下。所提出的工作还将导致通过李群（例如矩阵）中的对称性进行数据同步的新算法，以及同步问题只能近似解决的分析。这种结合束理论和切赫上同理论的纤维束工具的新颖适应性，将在纯数学和计算数学的前沿领域带来新的挑战和应用机会。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Modern technologies, like cellular devices and other sophisticated sensors, have made possible the collection of large volumes of data. In many applications the resulting data may arrive from different sources and at different times, or the amount of information is so large that it is impossible to analyze or store in a single computer. Examples of such applications include analytics in the cloud, or inference from databases distributed across a network of devices for privacy and/or security reasons.The need to analyze this type of data has prompted a growing class of mathematical, statistical and computational challenges,where distributed measurements need to be assembled to draw conclusions about a system. To date most research has focused on the computational question of efficiently and accurately synchronizing/aligning distributed data,but less is known about how to address the mathematical impediments to finding such solutions. This is the knowledge gap this project seeks to address. Specifically, the investigator will developthe theoretical, mathematical and algorithmic foundations needed to learn from distributed data,even when total synchronization is not possible. The tools developed in this project will advance pure and computational mathematics, and have the potential to be applied in areas such as cloud computing, distributed data visualization and sensor fusion. The impact of this research will be further amplified by its inclusion into novel educational materials developed by the investigator,as well as in the training of computational scientists and mathematicians.The main theme in this project is the adaptation of tools from classical algebraic topology -- the branch of mathematics concerned with the shape of abstract objects and how local constructions interact -- to the problem of learning from distributed data. Specifically, the research funded by this award seeks to: (1) leverage ideas from sheaf theory in order to develop algorithms capable of estimating, from data, the topological obstructions to the global synchronization of distributed measurements; and, (2) to utilize tools from the theory of fiber bundles to compute consistent assemblages of local data,even in the presence of non-trivial obstructions. The proposed work will also lead to novel algorithms for datasynchronization via symmetries in Lie (e.g., matrix) groups, and analyses where the synchronization problem can be solved only approximately. This novel adaptation of tools from fiber bundles in conjunction with sheaf theory and Cech cohomologywill present new challenges and application opportunities at the bleeding edge of pure and computational mathematics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.