High Dimensional Data Representations: Compressed Sensing, Randomized Row-Action Methods, and Quantization

该奖项将支持对高维问题中信号表示和信号重建的数学方法的研究。该项目研究随机行作用方法作为高维信号恢复问题的工具。行操作算法通常非常适合于大型问题，因为它们涉及在高维上很好缩放的低复杂度迭代。此外，行操作方法的在线特性适合于流式数据应用程序，其中一个顺序地获得对整个系统各个部分的访问。随机化是使行作用方法产生可证明的快速和准确结果的必要工具。该项目将研究高维应用的随机行作用方法，如从帧系数中重建信号、从量化样本中重建信号和稀疏近似。该项目还研究了如何在高维给物理设备、计算资源和数据存储带来巨大负担的情况下对信息进行数字编码的补充问题。该项目分析和设计了针对有限帧、压缩感知问题和非同步环境下的超宽带交错采样的高效、鲁棒的模数转换算法。现代数据集的庞大规模对数据的分析、数字化、处理和表示方式提出了根本性的挑战。不同的问题，如计算机断层扫描、高光谱成像、雷达和相位检索，可能涉及到如此高维的信号类，以至于经典设计的方法变得不切实际。该奖项将支持数学技术的发展，为新兴的高维问题提供数字信号表示和信号重建算法。高维数据的挑战需要超越线性信号表示的方法，并且能够有效地利用非线性结构化信号类并利用非常适度的过采样。典型的场景发生在高维数据中，例如流式高清视频，这些数据具有本质上较低维的信息内容，例如相对较少的移动形状。该项目的分析和算法将广泛应用于现代信号处理技术，如超宽带通信、压缩感知问题、一致重建、模数转换和sigma-delta量化。该奖项将支持研究生培训，研究生将参与该项目。

This award will support research on mathematical methods for signal representation and signal reconstruction in high dimensional problems. The project investigates randomized row-action methods as a tool for high dimensional signal recovery problems. Row-action algorithms are often well suited for large problems since they involve low complexity iterations that scale well in high dimensions. Moreover, the online nature of row-action methods is suitable for streaming data applications where one sequentially obtains access to individual parts of the entire system. Randomization is an essential tool for enabling row-action methods to yield provably fast and accurate results. The project will investigate randomized row-action methods for high dimensional applications such as signal reconstruction from frame coefficients, reconstruction from quantized samples, and sparse approximation. The project also studies the complementary problem of how to digitally encode information when high dimensionality places an enormous burden on physical devices, computational resources, and data storage. The project analyzes and designs efficient and robust analog-to-digital conversion algorithms for finite frames, compressed sensing problems, and ultra wideband interleaved sampling in non-synchronized environments. The enormous size of modern data sets poses fundamental challenges to the ways in which data is analyzed, digitized, processed, and represented. Disparate problems such as computerized tomography, hyperspectral imaging, radar, and phase retrieval can involve signal classes that are so high dimensional that classically designed methods become impractical. This award will support the development of mathematical techniques to provide digital signal representations and signal reconstruction algorithms for emerging classes of high dimensional problems. The challenges of high dimensional data require approaches that go beyond linear signal representations and which are able to efficiently take advantage of nonlinearly structured signal classes and capitalize on very modest oversampling. A typical scenario occurs for high dimensional data such as streaming HD video that have information content of intrinsically lower dimension such as relatively few moving shapes. The analysis and algorithms in this project will apply broadly to modern signal processing techniques such as ultra wideband communications, compressed sensing problems, consistent reconstruction, analog-to-digital conversion and sigma-delta quantization. The award will support graduate student training, and graduate students will be involved in the project.