ATD: Frame-Theoretic Algorithms for Smart Sensing

研究人员设计的显式传感矩阵与超越最先进水平的信号处理算法协调一致。它们朝着压缩感知和框架理论中几个最重要的公开问题的解决而前进：(A)找到具有受限等距性质(RIP)的矩阵的确定性结构；(B)构造实的和复的等角紧框架(ETF)，包括Zauner猜想存在的最大ETF；(C)设计快速可靠的相位恢复算法；(D)证明Wright猜想关于最小注射强度测量次数的最新改进；(E)Paulsen问题：如何投影到单位范数紧框架(UNTF)的流形上。阈值检测需要高质量的检测和有效的信号处理算法。大多数现代威胁都有模糊的信号，这给情报机构留下了两个主要问题。首先，你找不到你不想要的东西。也就是说，为了检测到威胁，您必须首先充分感知感兴趣的区域。正因为如此，部分工作集中在将提高传感能力的新数学上。这有一个不幸的副作用，即增加了现有的消防数据，加剧了第二个主要问题：“大海捞针”。眼下，分析师必须费力地处理难以想象的庞大数据集，才能提取出可采取行动的情报。因此，剩下的工作重点是处理大数据集的新数学方法。

The investigators design explicit sensing matrices in concert with signal processing algorithms that outperform the state of the art. They progress towards solutions of several of the most significant open problems in compressed sensing and frame theory: (a) to find deterministic constructions of matrices with the restricted isometry property (RIP); (b) to construct real and complex equiangular tight frames (ETFs), including the maximal ETFs conjectured to exist by Zauner; (c) to design fast and reliable algorithms for phase retrieval;(d) to prove recent refinements of Wright's conjecture regarding the minimal number of injective intensity measurements; (e) the Paulsen problem: How to project onto the manifold of unit norm tight frames (UNTFs).Threat detection requires high-quality sensing and effective signal processing algorithms. Most modern threats have faint signatures, leaving intelligence agencies with two main problems. First, you can't find what you don't seek. That is, in order to detect a threat, you must first adequately sense the region of interest. Because of this, part of the work focuses on new mathematics that will improve sensing capabilities. This has the unfortunate side-effect of adding to the existing fire hose of data, exacerbating the second main problem: "finding a needle in a haystack." At the moment, analysts must wade through unimaginably large data sets in order to extract actionable intelligence. As such, the remainder of the work focuses on new mathematical ways of processing big data sets.