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Structured Random Matrices and Graphs in Signal Processing

信号处理中的结构化随机矩阵和图

基本信息

批准号：
1909457
负责人：
Deanna Needell
金额：
$ 10.69万
依托单位：
University of California-Los Angeles
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-15 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1909457&HistoricalAwards=false
关键词：
Structured Random Matrices Graphs Signal

项目摘要

Recent advances in science and technology allow us to collect and require us to process large volumes of data of different types, ranging from routine user information collected by electronic devises, such as geotags, to more structurally complicated data obtained as a result of scientific experiments and biomedical imaging. Solving the associated signal processing problems requires an understanding of the underlying structure of data and finding its efficient representations. Frames proved to be a powerful tool in signal processing, providing redundant representations of signals that are useful for many applications. In return, signal processing problems motivate the study of frame properties. The primary objective of the project is to study connections between frame theory, with focus on time-frequency analysis, and signal processing problems, such as the phase retrieval problem, where a signal is reconstructed from intensity measurements with respect to a frame. More precisely, the investigator aims to study geometric properties of Gabor frames and their role in analysis of the phase retrieval problem with time-frequency structured measurements. Gabor frames naturally arise in such applications as speech recognition, so obtaining results on phase retrieval problem with time-frequency structured measurements would lead to advances in speech recognition technology and provable guarantees for existing ad hoc methods. In many signal processing applications, including social, transport, and epidemiology networks, data is naturally associated with the vertices of a weighted graph that represents the relations between data units. Processing of such data requires the development of different signal processing tools that would take the underlying graph structure into account. Another aim of the project is to extend classical discrete signal processing tools and concepts to signals defined on graphs. This would allow to solve classical signal processing problems in this generalized setting and advance in such important applications as weather and traffic prediction and brain imaging.The project is centered around three main objectives that are closely connected to each other, each addressing an important problem in the areas of random matrix theory, Gabor analysis, and signal processing. The first research direction is devoted to the study of geometric properties of frames. While properties of random Gaussian frames with independent vectors are sufficiently well-studied, very little is known about structured frames that are relevant for signal processing applications. This motivates the study of structured application relevant frames, such as Gabor frames. The investigator will focus on such properties of Gabor frames as optimal frame bounds and frame order statistics, with the ultimate goal of showing that Gabor frames have properties that are essentially optimal for applications. The study of the optimal frame bounds would allow to establish robustness of the signal reconstruction from its frame coefficients, justifying the use of the frame representation of a signal in practical applications. In the case of Gabor frames, the frame bounds depend not only on the cardinality of the frame set, but also on its structure. In the second research direction, the investigator aims to use studied properties of frames to analyze the phase retrieval problem. Even though there are many results on phase retrieval obtained for Gaussian measurements, the case of structured frames remains wide open. The main reason for this is that geometric properties of such frames are not yet fully understood. Getting new insights into the properties of Gabor frames would lead to a substantial progress in the development of the phase retrieval problem. In particular, obtaining uniform bounds on frame order statistics would imply invertibility and stability of the phase retrieval problem with Gabor frames. The third research direction of the project focuses on various aspects of time-frequency analysis for graph-based signals. The two main goals are to describe graphs for which Gabor frames have properties similar to the classical case, and to study properties of graph Gabor frames, depending on the underlying graph structure. To achieve the latter goal, the investigator aims to use random graph theory. The generalization of such cornerstone concepts as uncertainty principle and restricted isometry property would allow to develop Gabor analysis toolbox for graph-based signals and approach classical signal processing problems in this more general setting.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.

科学和技术的最新进展使我们能够收集并要求我们处理大量不同类型的数据，从地理标签等电子装置收集的日常用户信息到科学实验和生物医学成像获得的结构更复杂的数据。解决相关的信号处理问题需要理解数据的底层结构并找到其有效表示。事实证明，框架是信号处理中的一个强大工具，它提供了对许多应用有用的信号的冗余表示。反过来，信号处理问题激发了对帧特性的研究。该项目的主要目标是研究框架理论之间的联系，重点是时间-频率分析和信号处理问题，如相位恢复问题，其中信号从相对于框架的强度测量中重建。更确切地说，研究者的目的是研究几何性质的Gabor框架和它们的作用，在分析的相位恢复问题的时频结构化测量。Gabor框架在语音识别等应用中很自然地出现，因此，利用时频结构化测量获得相位恢复问题的结果将导致语音识别技术的进步和现有ad hoc方法的可证明保证。在许多信号处理应用中，包括社交网络、交通网络和流行病学网络，数据自然地与表示数据单元之间关系的加权图的顶点相关联。处理这些数据需要开发不同的信号处理工具，这些工具将考虑到底层的图形结构。该项目的另一个目的是将经典的离散信号处理工具和概念扩展到图上定义的信号。这将有助于解决在这种广义环境下的经典信号处理问题，并在天气和交通预测以及脑成像等重要应用中取得进展。该项目围绕着三个相互紧密联系的主要目标，每个目标都解决了随机矩阵理论，Gabor分析和信号处理领域的一个重要问题。第一个研究方向致力于研究框架的几何性质。虽然具有独立向量的随机高斯框架的性质已经得到了充分的研究，但对与信号处理应用相关的结构化框架知之甚少。这激发了结构化应用相关框架的研究，例如Gabor框架。调查人员将专注于这样的属性的Gabor帧作为最佳帧边界和帧顺序统计，最终的目标是显示，Gabor帧的属性，基本上是最佳的应用程序。最佳帧边界的研究将允许从其帧系数建立信号重构的鲁棒性，证明在实际应用中使用信号的帧表示是合理的。在Gabor框架的情况下，框架边界不仅取决于框架集的基数，而且还取决于它的结构。在第二个研究方向中，研究者的目的是利用研究的帧的属性来分析相位恢复问题。即使有许多结果的相位恢复高斯测量，结构化帧的情况下仍然是开放的。其主要原因是这种框架的几何特性尚未完全理解。对Gabor框架性质的深入了解将使相位恢复问题的发展取得实质性进展。特别是，获得统一的帧顺序统计量的界限将意味着可逆性和稳定性的相位恢复问题的Gabor框架。该项目的第三个研究方向侧重于基于图形的信号的时频分析的各个方面。两个主要目标是描述图的Gabor框架具有类似的经典情况下的属性，并研究图形的Gabor框架的属性，这取决于底层的图形结构。为了实现后一个目标，研究者的目标是使用随机图论。不确定性原理和限制等距属性等基本概念的推广将允许开发基于图形的信号的Gabor分析工具箱，并在此更一般的设置中接近经典信号处理问题。该奖项反映了NSF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。