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

AF：小：用于局部到全局推理的捆绑理论方法

• 批准号: 2006661
• 负责人: Jose Perea
• 金额: $ 35.08万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2006661&HistoricalAwards=false
• 关键词: AF; Small; Bundle; theoretic; methods

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.

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，产生的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。此类应用的示例包括云中的分析，或出于隐私和/或安全原因从分布在设备网络中的数据库进行推断。分析此类数据的需求引发了越来越多的数学、统计和计算挑战，其中需要组装分布式测量以得出有关系统的结论。到目前为止，大多数研究都集中在有效和准确地同步/对齐分布式数据的计算问题，但很少有人知道如何解决数学障碍，找到这样的解决方案。这就是这个项目试图解决的知识差距。具体来说，研究人员将开发从分布式数据中学习所需的理论，数学和算法基础，即使完全同步是不可能的。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。这项研究的影响将通过将其纳入研究者开发的新教材而进一步扩大，以及在培训计算科学家和数学家。在这个项目的主题是适应工具从经典代数拓扑-数学的分支有关的形状的抽象对象和如何当地建设相互作用-to the problem问题of learning学习from distributed分布式data数据.具体而言，该奖项资助的研究旨在：（1）利用层理论的思想，以开发能够从数据中估计分布式测量的全局同步的拓扑障碍的算法;以及（2）利用纤维束理论的工具来计算本地数据的一致集合，即使存在非平凡的障碍。所提出的工作还将导致通过Lie中的对称性进行数据化的新算法（例如，矩阵）组，并分析同步问题只能近似解决。这一新的适应工具从纤维束结合层理论和切赫cohomologywill提出新的挑战和应用机会的出血边缘的纯数学和计算mathematics.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

项目成果

Toroidal Coordinates: Decorrelating Circular Coordinates with Lattice Reduction

环形坐​​标：通过晶格缩减去关联圆坐标

• DOI: 10.4230/lipics.socg.2023.57
• 发表时间: 2023
• 期刊: Leibniz international proceedings in informatics
• 作者: Scoccola, Luis;Gakhar, Hitesh;Bush, Johnathan;Schonsheck, Nikolas;Rask, Tatum;Zhou, Ling;Perea, Jose A.
• 通讯作者: Perea, Jose A.

DREiMac: Dimensionality Reduction with Eilenberg-MacLane Coordinates

DREiMac：使用 Eilenberg-MacLane 坐标进行降维

• DOI: 10.21105/joss.05791
• 发表时间: 2023
• 期刊: Journal of Open Source Software
• 作者: Perea, Jose A.;Scoccola, Luis;Tralie, Christopher J.
• 通讯作者: Tralie, Christopher J.

Topological Data Analysis of Electroencephalogram Signals for Pediatric Obstructive Sleep Apnea

小儿阻塞性睡眠呼吸暂停脑电图信号的拓扑数据分析

• DOI: 10.1109/embc40787.2023.10340674
• 发表时间: 2023
• 期刊: Annual International Conference of the IEEE Engineering in Medicine and Biology Society
• 作者: Manjunath, Shashank;Perea, Jose A.;Sathyanarayana, Aarti
• 通讯作者: Sathyanarayana, Aarti

Persistable: persistent and stable clustering

• DOI: 10.21105/joss.05022
• 发表时间: 2023-03
• 期刊: J. Open Source Softw.
• 作者: Luis Scoccola;Alexander Rolle

Sliding window persistence of quasiperiodic functions

• DOI: 10.1007/s41468-023-00136-7
• 发表时间: 2021-03
• 期刊: Journal of Applied and Computational Topology
• 作者: H. Gakhar;Jose A. Perea