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

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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.