Applied Harmonic Analysis to Non-Convex Optimizations and Nonlinear Matrix Analysis

该项目在促进教学、培训和学习的同时，促进了对应用调和分析的科学理解。研究人员研究了两组问题，每一组都利用了数学和工程中表示的冗余性。第一种方法的一个例子是，当固定源不相关且并非所有传感器都从给定源接收信息时，从多个传感器阵列同时估计固定源的位置及其独立信号的内容。从数学上讲，这归结为从半正定协方差矩阵中提取信息，该矩阵将来自信号源的信号与传感器接收的信号联系起来。未知盲区的存在使这一问题具有挑战性。第一个重点是关于半正定有限迹算子的类。其目的是寻找将这类算子分解成使给定准则最小化的一阶算子的和。这个问题的困难在于秩1也是半正定的条件。事实证明，这个问题与Feichtinger在分析调制空间中具有核的紧算子时的一个公开问题有关。此外，这个问题与稀疏矩阵分解理论和阵列信号处理有很强的联系。第二个重点是关于低秩半正定矩阵类的分析和优化。特别是，这一推力延续了研究人员先前在相位恢复问题和量子态层析问题上的工作。在整个程序中使用了Lipschitz分析和非凸优化的工具。研究生参与了这项研究。此外，研究人员还通过与行业和政府实验室的联系，为他们提供具有全球竞争力的STEM劳动力培训。该项目加强了与工业界现有的伙伴关系，同时提供了探索现实世界应用的数学和为现有问题创造新的解决方案的机会。本文提出的第一个问题是关于H.Feichtinger在2004年提出的问题：一个核在第一个调制空间中的正半定迹类积分算子的特征分解是否收敛于一个更强的调制空间？事实证明，这个问题有一个否定的答案；然而，它自然提出了这样一个问题：这些运算符的不同分解(不一定是特征分解)是否在如此强烈的意义上收敛。类似的分解问题出现在盲源分离的上下文中。考虑由许多传感器(例如，天线或麦克风)和去相关的广义固定发射源组成的系统。混合环境有盲区，因此并不是所有的传感器都从给定源接收信息。问题是仅基于半正定协方差矩阵同时估计源和源信号的位置。未知盲点的存在使这一问题具有挑战性。该项目的第二个问题涉及量子状态层析成像和相位恢复。具体地说，它要求从具有固定埃尔米特矩阵集的内积估计低秩正半正定单位迹矩阵。本项目主要研究矩阵恢复的同伦方法类。研究生参与研究。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project advances scientific understanding in applied harmonic analysis while promoting teaching, training and learning. The investigator studies two sets of problems, each exploiting redundancy of representation in mathematics and engineering. An example of the first is the problem of estimating simultaneously, from an array of many sensors, the locations of stationary sources and the content of their separate signals, when the sources are not correlated and not all sensors receive information from a given source. Mathematically, this comes down to extracting information from a positive semi-definite covariance matrix that relates signals from sources to receptions by sensors. The existence of unknown blind spots makes this problem challenging. The first thrust concerns the class of positive semi-definite finite trace operators. The aim is to look for decompositions of such operators into sums of rank-one operators that minimize a given criterion. What makes this problem hard is the condition that the rank-ones are also positive semi-definite. It turns out this problem is connected to an open question of Feichtinger in analysis of compact operators with kernels in a modulation space. Additionally, the problem has strong connections to the theory of sparse matrix decomposition, and to array signal processing. The second thrust is related to analysis and optimizations on classes of low-rank positive semi-definite matrices. In particular, this thrust continues the investigator's previous work on the phase retrieval problem and the quantum state tomography problem. Tools from Lipschitz analysis and non-convex optimizations are used throughout this program. Graduate students participate in the research. In addition, the investigator is training them for a globally competitive STEM workforce through his contacts with industry and government labs. The project strengthens existing partnerships with industry while offering opportunities to explore mathematics of real-world applications and to create novel solutions to existing problems. Undergraduate students are encouraged to enter this area of research by participating in existing opportunities under the umbrellas of Research Experience for Undergraduates or Research Interaction Teams.The first problem proposed here relates to the question H. Feichtinger asked in 2004: does the eigen-decomposition of a positive semi-definite trace-class integral operator with kernel in the first modulation space converge in a stronger modulation space-sense? It turns out this question has a negative answer; however, it naturally raises the question of whether a different decomposition of such operators (not necessarily the eigen-decomposition) converges in such a stronger sense. A similar decomposition problem appears in the context of blind source separation. Consider a system composed of many sensors (e.g., antennas, or microphones) and decorrelated wide-sense stationary transmitting sources. The mixing environment has blind spots so that not all sensors receive information from a given source. The problem is to estimate simultaneously location of sources and the source signals, based only on the positive semi-definite covariance matrix. The existence of unknown blind spots is what makes this problem challenging. The second problem of this project refers to quantum state tomography and phase retrieval. Specifically, it asks to estimate low-rank positive semi-definite unit trace matrices from inner products with a fixed set of Hermitian matrices. The project focuses on the class of homotopy methods for matrix recovery. Graduate students participate in the research.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.