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，促进了对应用谐波分析的科学理解。研究者研究了两组问题，每组问题都利用了数学和工程中的表示冗余。第一个问题的一个例子是，当源不相关且并非所有传感器都从给定源接收信息时，从许多传感器阵列同时估计固定源的位置及其单独信号的内容的问题。从数学上讲，这可以归结为从一个正的半定协方差矩阵中提取信息，该矩阵将信号从源与传感器的接收联系起来。未知盲点的存在使得这个问题具有挑战性。第一部分涉及一类正半确定有限跟踪算子。目的是寻找这些算子的分解成使给定准则最小化的第一阶算子的和。使这个问题变得困难的是秩1也是正半定的条件。结果表明，这一问题与Feichtinger在分析调制空间中带核紧算子的开放性问题有关。此外，该问题与稀疏矩阵分解理论和阵列信号处理有很强的联系。第二个推力与低秩半正定矩阵类的分析和优化有关。特别是，这一推力继续了研究者之前在相位恢复问题和量子态层析问题上的工作。Lipschitz分析和非凸优化工具在整个程序中使用。研究生参与研究。此外，研究人员正在通过与行业和政府实验室的联系，培训他们成为具有全球竞争力的STEM劳动力。该项目加强了与工业界的现有合作伙伴关系，同时为探索现实世界的数学应用提供了机会，并为现有问题创造了新的解决方案。鼓励本科生通过参与本科生研究经验或研究互动团队的现有机会进入这一研究领域。这里提出的第一个问题与H. Feichtinger在2004年提出的问题有关：在第一调制空间中具有核的正半定迹类积分算子的特征分解是否收敛于更强的调制空间感？这个问题的答案是否定的；然而，它自然地提出了这样一个问题：这些算子的不同分解（不一定是特征分解）是否在如此强的意义上收敛。在盲源分离中也出现了类似的分解问题。考虑一个由许多传感器（例如，天线或麦克风）和去相关的广域静止发射源组成的系统。混合环境存在盲点，因此并非所有传感器都能从给定源接收信息。问题是同时估计源和源信号的位置，仅基于正半定协方差矩阵。未知盲点的存在使得这个问题具有挑战性。本课题的第二个问题是量子态层析和相位恢复。具体来说，它要求从一组固定的厄米矩阵的内积估计低秩正半定单位迹矩阵。本项目主要研究一类矩阵恢复的同伦方法。研究生参与研究。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.