Structured Random Matrices and Graphs in Signal Processing

最近的科学和技术进步使我们能够收集并要求我们处理大量不同类型的数据，从通过地理标签等电子设备收集的常规用户信息，到通过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了解数据的基本结构并找到其有效的表示法。事实证明，帧是信号处理中的一个强大工具，它提供了对许多应用程序有用的信号的冗余表示。反过来，信号处理问题也推动了对帧特性的研究。该项目的主要目标是研究以时频分析为重点的框架理论与信号处理问题之间的联系，例如相位恢复问题，在该问题中，根据相对于帧的强度测量来重建信号。更准确地说，研究人员的目标是研究Gabor帧的几何性质，以及它们在分析具有时频结构测量的相位恢复问题中的作用。Gabor帧在语音识别等应用中是自然存在的，因此利用时频结构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，并为现有的自组织方法提供可证明的保证。在许多信号处理应用中，包括社会、交通和流行病学网络，数据自然与表示数据单元之间关系的加权图的顶点相关联。这种数据的处理需要开发不同的信号处理工具，这些工具将考虑到基本的图形结构。该项目的另一个目标是将经典的离散信号处理工具和概念扩展到在图形上定义的信号。这将允许在这种普遍的背景下解决经典的信号处理问题，并在天气和交通预测以及脑成像等重要应用中取得进展。该项目围绕三个相互紧密联系的主要目标展开，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。第一个研究方向是对框架几何性质的研究。虽然具有独立向量的随机高斯帧的性质已被充分研究，但对于与信号处理应用相关的结构化帧却知之甚少。这激发了对结构化应用相关框架的研究，例如Gabor框架。研究人员将重点研究Gabor帧的性质，如最优帧界限和帧顺序统计量，最终目标是证明Gabor帧具有本质上是最优的应用性质。对最佳帧界限的研究将允许根据其帧系数建立信号重构的稳健性，从而证明在实际应用中使用信号的帧表示是合理的。在Gabor帧的情况下，帧的界限不仅取决于帧集合的基数，而且取决于其结构。在第二个研究方向中，研究者的目标是利用已学习的帧的性质来分析相位恢复问题。尽管在高斯量测的相位恢复方面已经取得了很多结果，但结构框架的情况仍然很难确定。其主要原因是对这类框架的几何性质还没有完全了解。对Gabor帧的性质有了新的认识，将在相位恢复问题的发展方面取得实质性的进展。特别是，获得一致的帧顺序统计量将意味着Gabor帧相位恢复问题的可逆性和稳定性。项目的第三个研究方向集中在基于图的信号的时频分析的各个方面。两个主要目标是描述Gabor框架具有类似于经典情况的性质的图，以及研究依赖于底层图结构的图Gabor框架的性质。为了实现后一种目标，研究者的目标是使用随机图理论。不确定原理和受限等距性质等基石概念的推广将允许开发基于图形的信号的Gabor分析工具箱，并在这种更一般的环境中处理经典信号处理问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.