Sampling and quantization theorems for modern data acquisition

该奖项支持首席研究员在信号处理一般领域的研究项目，例如，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。为了实现准确的信号恢复和优化后续使用，设计良好的测量和数字化过程至关重要。理解它们的理论性质、性能保证和局限性也很重要。在实际限制下，我们在理论上不仅研究如何进行测量，而且研究如何将其数字化（即，如何将其转换为比特流并随后压缩）。首席研究员将专注于非经典信号模型（包括稀疏向量和低秩矩阵），由于其与现代应用的相关性，近年来其重要性不断增加。该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-所有这些都是在非线性算法的信号重建的现代设置中。例如，它关注使用压缩感知技术测量的近似稀疏信号，通过随机采样其条目测量的低秩矩阵，以及由与其他向量的内积的大小组成的无相位测量的向量。简而言之，首席研究员为数字世界寻找采样和量化定理。这需要开发和使用来自各个数学领域的工具。由于与本研究相关的测量模型严重依赖于随机性，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。在制定冗余测量的量化方法时，将与框架理论和噪声整形量化的数学文献建立联系并加以利用。在本研究中提出的编码（压缩）算法广泛使用随机性，在这里，首席研究员期望与随机数值线性代数文献的联系。在分析新的重建算法时，他将采用并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.