Phaseless Reconstruction and Geometric Analysis of Frames

研究人员研究了两个问题，每个问题都利用了数学和工程中表示的冗余性，并开发了从非线性处理方案中恢复信号的新方法。第一个问题涉及从冗余线性表示的幅度重构信号(所谓的相位恢复问题)。第二个问题涉及对框架的几何分析以及与数学中的深层问题(如Kadison-Singer问题)的联系。这种分析导致了更快的方法，在X射线结晶学、光纤数据通信和语音处理的应用中提供了更好的重建信号的质量和分辨率。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，为具有全球竞争力的STEM劳动力进行培训。研究人员研究了两个问题，每个问题都利用了数学和工程中表示的冗余性。第一种方法导致从非线性处理方案中恢复信号的新方法。最近，在更大的嵌入空间中，有两个深远的发现将非线性信息(帧系数的大小)与某些标量积联系起来。这样，恢复信号的原始问题--基本上是非线性的--被重塑为线性重建问题和一阶逼近问题。当线性冗余表示与群表示(例如Weyl-Heisenberg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矢量来表示，而是由更大维类希尔伯特-施密特空间中的算符来表示，类似于量子态理论。这里开发的方法借鉴了调和分析、算子理论和代数几何等广泛的数学领域。反过来，这种方法允许稳定和高效的解决方案与电子工程领域相关，包括阵列信号处理、语音处理、量子计算和X射线结晶学。第二个问题将Kadison-Singer问题的解在框架理论中向不同的方向展开。具体地说，问题是将帧“细化”到保留帧的子集，这些子集的密度任意接近1，这是与Riesz基关联的临界密度。这样的结果属于描述框架集几何的更大的结果主体。在所有这些问题中，统一的概念是表示的冗余和原子分解。

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.