Phaseless Reconstruction and Geometric Analysis of Frames

研究者研究了两个问题，每个问题都利用了数学和工程中的冗余表示，并开发了从非线性处理方案中恢复信号的新方法。第一个问题与从冗余线性表示的幅值重建信号有关（即所谓的相位恢复问题）。第二个问题涉及到框架的几何分析，以及与数学中的深层问题（如卡迪逊-辛格问题）的联系。这种分析导致更快的方法，在x射线晶体学、光纤数据通信和语音处理等应用中提供更好的质量和分辨率的重建信号。参与该项目的本科生和研究生通过学习开发新的数学工具来解决现实世界的问题，培养具有全球竞争力的STEM劳动力。研究者研究了两个问题，每个问题都利用了数学和工程中的冗余表示。首先引出了从非线性处理方案中恢复信号的新方法。最近有两个影响深远的发现，将非线性信息（框架系数的大小）与较大嵌入空间中的某些标量积联系起来。通过这种方式，原始的信号恢复问题，本质上是非线性的，被重新转化为一个线性重建问题，加上一个秩一近似问题。当线性冗余表示与群表示（如Weyl-Heisenberg变换或加窗傅立叶变换）相关联时，相关张量算子继承这种不变性。因此，快速（非线性）重建算法是可能的。这种方法提出了一种新的信号表示模型，其中信号不是简单地由希尔伯特空间中的向量表示，而是由更大维度的希尔伯特-施密特空间中的算子表示，类似于量子态理论。这里开发的方法借鉴了广泛的数学领域，如谐波分析、算子理论和代数几何。反过来，这种方法可以为阵列信号处理、语音处理、量子计算和x射线晶体学等电气工程领域提供稳定高效的解决方案。第二个问题在框架理论的另一个方向上扩展了卡迪逊-辛格问题的解。具体来说，问题是将帧“细化”为子集，这些子集仍然是帧，并且密度任意接近于1，即与Riesz基相关的临界密度。这样的结果属于描述框架集几何的一个更大的结果体。所有这些问题的统一概念是表示冗余和原子分解。

The investigator studies two problems, each exploiting redundancy of representations in mathematics and engineering, and develops new methods to recover a signal from a nonlinear processing scheme. The first problem is related to signal reconstruction from magnitudes of a redundant linear representation (the so-called phase retrieval problem). The second problem involves a geometric analysis of frames and connections to deep problems in mathematics (such as the Kadison-Singer problem). This analysis leads to faster methods that offer better quality and resolution of the reconstructed signals in applications from X-ray crystallography, data communication on fiber optics, and speech processing. Undergraduate and graduate students involved in this project are trained for a globally competitive STEM workforce by learning to develop new mathematical tools to solve real-world problems. The investigator studies two problems, each exploiting redundancy of representations in mathematics and engineering. The first leads to new methods to recover a signal from a nonlinear processing scheme. Recently two far-reaching discoveries have been made that connect the nonlinear information (magnitudes of frame coefficients) to certain scalar products in larger embedding spaces. This way the original problem of recovering a signal, which is fundamentally nonlinear, is recast into a linear reconstruction problem coupled with a rank-one approximation problem. When the linear redundant representation is associated with a group representation (such as Weyl-Heisenberg, or windowed Fourier transform), then the relevant tensor operators inherit this invariance property. Thus a fast (nonlinear) reconstruction algorithm is possible. This approach suggests a new signal representation model, where signals are not represented simply by vectors in a Hilbert space, but rather by operators in a larger dimensional Hilbert-Schmidt-like space, similar to the quantum state theory. The methods developed here borrow from a wide range of mathematical areas such as harmonic analysis, operator theory, and algebraic geometry. In turn this approach allows for stable and efficient solutions relevant to areas of electrical engineering as diverse as array signal processing, speech processing, quantum computing, and X-ray crystallography. The second problem expands the solution of the Kadison-Singer problem in a different direction in frame theory. Specifically the issue is to "thin out" frames to subsets that remain frames and have density arbitrary close to one, the critical density associated to a Riesz basis. Such a result belongs to a larger body of results describing the geometry of frame sets. The unifying concept in all these problems is redundancy of representations and atomic decompositions.