Mathematical Foundation for Signal Processing on Spatially Distributed Networks

空间分布式数据网络由大量具有数据收集、处理和共享能力的协作代理组成。它已被广泛应用于（无线）传感器网络、智能电网以及其他工程和科学领域，因为它比将收集到的数据运送到中央处理设施的更传统的数据网络形式提供了一些实际优势。然而，数据处理任务的分布，如滤波、去噪和压缩，提出了一些新的挑战。近年来，数学理论与工程实践之间仍有一定的差距。研究者致力于在空间分布式数据网络上开发信号处理的数学基础，这对工程应用具有实际影响。研究生参与研究。研究生参与研究。空间分布式数据网络的拓扑结构用稀疏图来描述。空间分布式数据网络上的数据处理由网络中的代理执行；其中许多可以用良好定域的矩阵和积分算子来描述，原则上可以用它们来描述和全局优化网络行为。但这可能是一个不切实际的大型优化问题。首席研究员研究如何将一个大网络分成一组重叠的小网络，这些小网络的单独全局优化可以用来近似大网络的全局优化。通过这种方式，他的目标是为这种具有工程应用的网络上的信号处理建立数学基础。在这里，他强调了图上良好定域矩阵的谐波分析和数值分析的应用，稳定的空间分布数据网络的设计，空间分布数据网络中agent实现的快速算法，无相信号重建，以及图上信号处理的深度学习。研究生参与研究。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。

A spatially distributed data network consists of a large number of cooperative agents that have data collecting, processing, and sharing capabilities. It has been widely used in (wireless) sensor networks, smart power grids, and other engineering and science fields, because it offers some practical advantages over the more traditional form of data network that ships collected data to central processing facilities. However, distributing the data processing tasks, such as filtering, denoising, and compressing, raises some new challenges. While these have received much attention in recent years, there still is a gap between mathematical theory and engineering practice. The investigator works to develop a mathematical foundation for signal processing on spatially distributed data networks that has practical consequences for engineering applications. Graduate students participate in the research. Graduate students participate in the research.The topological structure of a spatially distributed data network is described by a sparse graph. Data processing on a spatially distributed data network is performed by agents in the network; many of them can be described by well-localized matrices and integral operators, in terms of which the network behavior can in principle be described and optimized globally. But this could be an impractically large optimization problem. The principal investigator studies how to split a large network into a set of overlapping smaller ones whose separate global optimizations can be used to approximate the global optimization of the large network. In this way, he aims to develop a mathematical foundation for signal processing on such networks that has engineering applications. Here he emphasizes applied harmonic analysis and numerical analysis of well-localized matrices on graphs, design of stable spatially distributed data networks, fast algorithms implemented by agents in spatially distributed data networks, phaseless signal reconstruction, and deep learning coupled to signal processing on graphs. Graduate students participate in the research.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.