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

将调和分析应用于非凸优化和非线性矩阵分析

基本信息

批准号：
1816608
负责人：
Radu Balan
金额：
$ 42.34万
依托单位：
University of Maryland, College Park
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-09-01 至 2022-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1816608&HistoricalAwards=false
关键词：
Applied Harmonic Analysis Non Convex

项目摘要

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.

该项目在促进教学，培训和学习的同时，在应用谐波分析中提高了科学理解。研究人员研究了两组问题，每组都利用了数学和工程学中表示的冗余。第一个示例是从许多传感器的数组，固定源的位置及其单独信号的内容同时估算的问题，当源不相关并且并非所有传感器都从给定源接收信息。从数学上讲，这取决于从正向半明确协方差矩阵中提取信息，该矩阵将信号从源到接收者的接收器。未知盲点的存在使这个问题具有挑战性。第一个推力涉及积极的半明确有限痕量操作员的类别。目的是将此类运营商的分解为最小化给定标准的排名一键运算符。使这个问题很难的原因是，排名也是积极的半明确者。事实证明，这个问题与Feichtinger的一个开放问题有关，用于分析与调制空间中内核的紧凑型操作员。此外，该问题与稀疏基质分解理论以及阵列信号处理具有牢固的联系。第二个推力与对低级别阳性半明确矩阵类别的分析和优化有关。特别是，这一推力继续了研究者先前在阶段检索问题和量子状态断层扫描问题上的工作。在整个程序中，都使用Lipschitz分析和非凸优化的工具。研究生参加研究。此外，调查员正在通过与行业和政府实验室的联系来培训他们的全球竞争性STEM劳动力。该项目加强了与行业的现有合作伙伴关系，同时提供了探索现实世界应用数学的机会，并为现有问题创建新颖的解决方案。鼓励本科生通过在研究经验的遮阳伞下参与现有机会的本科生或研究互动团队的现有机会。此处提出的第一个问题与H. Feichtinger在2004年提出的问题有关：是否在2004年提出的问题：是否对第一个模型中的强度半定型空间中的欧洲特征分类组成了一个积极的痕迹量化空间？事实证明，这个问题有负面答案。但是，它自然提出了这样一个问题，即此类操作员（不一定是特征分类）是否会在如此强大的意义上收敛。类似的分解问题出现在盲源分离的背景下。考虑一个由许多传感器（例如天线或麦克风）组成的系统，并与宽宽的固定传输源相关。混合环境具有盲点，因此并非所有传感器都从给定来源接收信息。问题在于仅基于正半准协方差矩阵，同时估计源和源信号的位置。未知盲点的存在是使这个问题具有挑战性的原因。该项目的第二个问题是指量子状态断层扫描和相位检索。具体而言，它要求从内部产物中估算具有固定的Hermitian矩阵的内部产物的低级阳性半明确单位痕量矩阵。该项目着重于用于矩阵恢复的同型方法类别。研究生参加了研究。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评论标准来评估值得支持的。