Applied Harmonic Analysis Methods for Non-Convex Optimizations and Low-Rank Matrix Analysis

该项目通过扩大应用调和分析在数学和应用方面的领域，促进了科学的进步。这项研究解决了目前超级计算机难以解决的复杂优化问题。这些项目拓宽了数学与工程和计算机科学之间的双向交流，同时促进了教学、培训和学习。PI将通过在行业和政府实验室的实习，为具有全球竞争力的STEM力量培养研究生。该项目增加了与产业界现有的伙伴关系，同时提供了探索现实世界应用程序的数学和为现有问题创造新的解决方案的机会。该项目包括两个方面。第一个推力发展了非凸优化的同伦方法。特别是，该项目将专注于低阶矩阵估计，如在相位恢复问题中的情况，以及二次分配问题，如在图匹配问题中。同伦方法通过一个连续参数来扩展原始搜索空间(相空间)，该参数在目标非凸目标函数和精心选择的凸罚项之间转换。在全局最优器处初始化路径跟踪器，并逐步向非凸目标函数的全局最优解进化。研究将获得该方法的最优性保证。第二个推论涉及低秩正半定矩阵的几何和泛函分析。该程序研究了这些度量空间的各种度量和Lipschitz嵌入。初步结果显示了关于这种半代数变体的丰富而复杂的度量集合。具体地说，一个这样的度量与非负一阶矩阵的和的最优展开有关。证明了这种分解与某些调制空间中带核的紧积分算子的分析直接相关。这些结果将扩展到有限维环境。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 promotes the progress of science by expanding the area of applied harmonic analysis both in mathematics and in applications. The research addresses complex optimization problems that are intractable by current super-computers. The projects broaden the two-way communication between mathematics on one side and engineering and computer science on the other side while promoting teaching, training, and learning. The PI will train graduate students for a globally competitive STEM force through internships in industry and government labs. The project increases the existing partnerships with industry while offering opportunities to explore mathematics of real-world applications and to create novel solutions to existing problems.The project contains two thrusts. The first thrust develops homotopic methods for non-convex optimizations. In particular, the project will focus on low-rank matrix estimation, such as the case in the phase retrieval problem, and quadratic assignment problems, as in graph matching problems. The homotopic method extends the original search space (the phase space) by one continuous parameter that trades between the target non-convex objective function and a carefully chosen convex penalty term. A path tracker is initialized at the global optimizer and gradually evolved towards the global optimum of the non-convex objective function. The research will obtain guarantees of optimality for this method. The second thrust relates to geometric and functional analysis of low-rank positive semi-definite matrices. The program studies various metrics and Lipschitz embeddings of these metric spaces. Preliminary results show a rich and complex collection of metrics on this semi-algebraic variety. In particular, one such measure is related to optimal expansions into sums of nonnegative rank-one matrices. It turns out this decomposition is directly related to the analysis of compact integral operators with kernels in certain modulation spaces. These results will be extended to finite-dimensional settings.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.