Sampling and quantization theorems for modern data acquisition

该奖项支持首席研究员在一般信号处理领域的研究计划，该计划本身涉及信息的电子传输：数字化、压缩、编码、解码等。现代传感器无处不在。它们测量感兴趣的信号，并以数字方式存储或传输大量数据。然后，复杂的计算机算法处理数据，恢复信号或执行其他任务。为了实现准确的数字信号恢复并优化后续使用，关键是要设计好测量和数字化过程。同样重要的是，了解它们的理论属性、性能保证和限制。在实际约束下，我们不仅应该从理论上调查应该如何进行测量，而且应该研究如何将它们数字化(即应该如何将它们转换为比特流并随后进行压缩)。首席调查员将专注于非经典信号模型(包括稀疏向量和低阶矩阵)，由于它们与现代应用程序的相关性，近年来它们的重要性一直在增加。本项目旨在了解结构化信号采样、数字化和压缩之间的相互作用-所有这些都是在通过非线性算法重建信号的现代背景下。例如，它侧重于使用压缩传感技术测量的近似稀疏信号、通过随机采样其条目来测量的低阶矩阵，以及其无相位测量由与其他矢量的内积的幅度组成的矢量。简而言之，首席调查者为数字世界寻求采样和量化定理。这需要开发和使用数学各个领域的工具。由于与本研究相关的测量模型严重依赖随机性，该项目必须开发和使用几何泛函分析和非渐近随机矩阵理论的方法。在冗余测量的量化方法的制定中，将建立和使用与框架理论和关于噪声整形量化的数学文献的联系。在这项研究中提出的编码(压缩)算法广泛使用随机性，在这里，首席调查者预计会与随机化的数值线性代数文献相联系。在分析新的重建算法时，他将使用和发展凸优化和数值分析的方法。

This award supports the research program of the Principal Investigator in the general area of signal processing, which concerns itself, for example, with the electronic transmission of information: digitization, compression, encoding, decoding, and the like. Modern sensors are ubiquitous. They measure signals of interest and digitally store or transmit large amounts of data. Sophisticated computer algorithms then process the data, to recover the signals or to perform other tasks. To allow accurate signal recovery and optimize subsequent use, it is critical that the measurement and digitization processes be well designed. It is also important that their theoretical properties, performance guarantees, and limitations be understood.  Under practical constraints, we theoretically investigate not only how measurements should be made, but also how they should be digitized (i.e., how they should be converted into bit-streams and subsequently compressed). The Principal Investigator will focus on nonclassical signal models (including sparse vectors and low-rank matrices) whose importance has been increasing in recent years due to their relevance to modern applications.This project aims to understand the interplay between sampling of structured signals, digitization, and compression---all in the modern setting of signal reconstruction by non-linear algorithms.  For example, it focuses on approximately sparse signals measured using compressed sensing techniques, low-rank matrices measured by randomly sampling their entries, and vectors whose phase-less measurements consist of the magnitudes of inner products with other vectors.  In short, the Principal Investigator seeks sampling and quantization theorems for the digital world. This requires developing and using tools from various areas of mathematics. As the measurement models related to this research rely heavily on randomness, the project must develop and use methods from geometric functional analysis and nonasymptotic random matrix theory. In the formulation of quantization methods for redundant measurements, connections with frame theory and the mathematical literature on noise-shaping quantization will be established and used. The encoding (compression) algorithms proposed in this research use randomness extensively, and here the Principal Investigator anticipates connections with the randomized numerical linear algebra literature. In analyzing new reconstruction algorithms, he will employ and develop methods from convex optimization and numerical analysis.